:: Some Isomorphisms Between Functor Categories
:: by Andrzej Trybulec
::
:: Received June 5, 1992
:: Copyright (c) 1992-2012 Association of Mizar Users


begin

definition
let A, B, C be non empty set ;
let f be Function of A,(Funcs (B,C));
:: original:uncurry
redefine func uncurry f ->  Function of [:A,B:],C;
coherence
 uncurry f is Function of [:A,B:],C
proofend;
end;

theorem Th1: :: ISOCAT_2:1
for A, B, C being non empty set
for f being Function of A,(Funcs (B,C)) holds curry (uncurry f) = f
proofend;

theorem Th2: :: ISOCAT_2:2
for A, B, C being non empty set
for f being Function of A,(Funcs (B,C))
for a being Element of A
for b being Element of B holds (uncurry f) . (a,b) = (f . a) . b
proofend;

theorem :: ISOCAT_2:3
canceled;

theorem :: ISOCAT_2:4
canceled;

theorem Th5: :: ISOCAT_2:5
for A being Category
for f being Morphism of A holds (id (cod f)) (*) f = f
proofend;

theorem Th6: :: ISOCAT_2:6
for A being Category
for f being Morphism of A holds f (*) (id (dom f)) = f
proofend;

theorem Th7: :: ISOCAT_2:7
for A, B being Category
for o being set holds
( o is Object of (Functors (A,B)) iff o is Functor of A,B )
proofend;

theorem Th8: :: ISOCAT_2:8
for A, B being Category
for f being Morphism of (Functors (A,B)) ex F1, F2 being Functor of A,B ex t being natural_transformation of F1,F2 st
( F1 is_naturally_transformable_to F2 & dom f = F1 & cod f = F2 & f = [[F1,F2],t] )
proofend;

begin

definition
let A, B be Category;
let a be Object of A;
funca |-> B ->  Functor of Functors (A,B),B means :Def1: :: ISOCAT_2:def 1
 for F1, F2 being Functor of A,B
for t being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 holds
it . [[F1,F2],t] = t . a;
existence
 ex b1 being Functor of Functors (A,B),B st
for F1, F2 being Functor of A,B
for t being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 holds
b1 . [[F1,F2],t] = t . a
proofend;
uniqueness
 for b1, b2 being Functor of Functors (A,B),B st ( for F1, F2 being Functor of A,B
for t being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 holds
b1 . [[F1,F2],t] = t . a ) & ( for F1, F2 being Functor of A,B
for t being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 holds
b2 . [[F1,F2],t] = t . a ) holds
b1 = b2
proofend;
end;

:: deftheorem Def1 defines |-> ISOCAT_2:def_1_:_
 for A, B being Category
for a being Object of A
for b4 being Functor of Functors (A,B),B holds
( b4 = a |-> B iff for F1, F2 being Functor of A,B
for t being natural_transformation of F1,F2 st F1 is_naturally_transformable_to F2 holds
b4 . [[F1,F2],t] = t . a );

theorem :: ISOCAT_2:9
for A being Category
for o, m being set holds Functors ((1Cat (o,m)),A) ~= A
proofend;

begin

theorem Th10: :: ISOCAT_2:10
for A, B, C being Category
for F being Functor of [:A,B:],C
for a being Object of A
for b being Object of B holds (F ?- a) . b = F . [a,b]
proofend;

theorem Th11: :: ISOCAT_2:11
for A, B being Category
for a1, a2 being Object of A
for b1, b2 being Object of B holds
( ( Hom (a1,a2) <> {} &amp; Hom (b1,b2) <> {} ) iff Hom ([a1,b1],[a2,b2]) <> {} )
proofend;

theorem Th12: :: ISOCAT_2:12
for A, B being Category
for a1, a2 being Object of A
for b1, b2 being Object of B st Hom ([a1,b1],[a2,b2]) <> {} holds
for f being Morphism of A
for g being Morphism of B holds
( [f,g] is Morphism of [a1,b1],[a2,b2] iff ( f is Morphism of a1,a2 &amp; g is Morphism of b1,b2 ) )
proofend;

theorem Th13: :: ISOCAT_2:13
for A, B, C being Category
for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2
for a being Object of A holds
( F1 ?- a is_naturally_transformable_to F2 ?- a &amp; (curry t) . a is natural_transformation of F1 ?- a,F2 ?- a )
proofend;

definition
let A, B, C be Category;
let F be Functor of [:A,B:],C;
let f be Morphism of A;
func curry (F,f) ->  Function of the carrier' of B, the carrier' of C equals :: ISOCAT_2:def 2
(curry F) . f;
coherence
 (curry F) . f is Function of the carrier' of B, the carrier' of C
proofend;
end;

:: deftheorem  defines curry ISOCAT_2:def_2_:_
 for A, B, C being Category
for F being Functor of [:A,B:],C
for f being Morphism of A holds curry (F,f) = (curry F) . f;

theorem Th14: :: ISOCAT_2:14
for A, B being Category
for a1, a2 being Object of A
for b1, b2 being Object of B
for f being Morphism of A
for g being Morphism of B st f in Hom (a1,a2) &amp; g in Hom (b1,b2) holds
[f,g] in Hom ([a1,b1],[a2,b2])
proofend;

theorem Th15: :: ISOCAT_2:15
for A, B, C being Category
for F being Functor of [:A,B:],C
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds
( F ?- a is_naturally_transformable_to F ?- b &amp; (curry (F,f)) * (IdMap B) is natural_transformation of F ?- a,F ?- b )
proofend;

definition
let A, B, C be Category;
let F be Functor of [:A,B:],C;
let f be Morphism of A;
funcF ?- f ->  natural_transformation of F ?- (dom f),F ?- (cod f) equals :: ISOCAT_2:def 3
(curry (F,f)) * (IdMap B);
coherence
 (curry (F,f)) * (IdMap B) is natural_transformation of F ?- (dom f),F ?- (cod f)
proofend;
correctness
;
end;

:: deftheorem  defines ?- ISOCAT_2:def_3_:_
 for A, B, C being Category
for F being Functor of [:A,B:],C
for f being Morphism of A holds F ?- f = (curry (F,f)) * (IdMap B);

theorem Th16: :: ISOCAT_2:16
for A, B, C being Category
for F being Functor of [:A,B:],C
for g being Morphism of A holds F ?- (dom g) is_naturally_transformable_to F ?- (cod g)
proofend;

theorem Th17: :: ISOCAT_2:17
for A, B, C being Category
for F being Functor of [:A,B:],C
for f being Morphism of A
for b being Object of B holds (F ?- f) . b = F . (f,(id b))
proofend;

theorem Th18: :: ISOCAT_2:18
for A, B, C being Category
for F being Functor of [:A,B:],C
for a being Object of A holds id (F ?- a) = F ?- (id a)
proofend;

theorem Th19: :: ISOCAT_2:19
for A, B, C being Category
for F being Functor of [:A,B:],C
for g, f being Morphism of A st dom g = cod f holds
for t being natural_transformation of F ?- (dom f),F ?- (dom g) st t = F ?- f holds
F ?- (g (*) f) = (F ?- g) `*` t
proofend;

definition
let A, B, C be Category;
let F be Functor of [:A,B:],C;
func export F ->  Functor of A, Functors (B,C) means :Def4: :: ISOCAT_2:def 4
 for f being Morphism of A holds it . f = [[(F ?- (dom f)),(F ?- (cod f))],(F ?- f)];
existence
 ex b1 being Functor of A, Functors (B,C) st
for f being Morphism of A holds b1 . f = [[(F ?- (dom f)),(F ?- (cod f))],(F ?- f)]
proofend;
uniqueness
 for b1, b2 being Functor of A, Functors (B,C) st ( for f being Morphism of A holds b1 . f = [[(F ?- (dom f)),(F ?- (cod f))],(F ?- f)] ) &amp; ( for f being Morphism of A holds b2 . f = [[(F ?- (dom f)),(F ?- (cod f))],(F ?- f)] ) holds
b1 = b2
proofend;
end;

:: deftheorem Def4 defines export ISOCAT_2:def_4_:_
 for A, B, C being Category
for F being Functor of [:A,B:],C
for b5 being Functor of A, Functors (B,C) holds
( b5 = export F iff for f being Morphism of A holds b5 . f = [[(F ?- (dom f)),(F ?- (cod f))],(F ?- f)] );

Lm1:  for A, B being Category
for F1, F2 being Functor of A,B st F1 is_transformable_to F2 holds
for t being transformation of F1,F2
for a being Object of A holds t . a in Hom ((F1 . a),(F2 . a))
proofend;

theorem Th20: :: ISOCAT_2:20
for A, B, C being Category
for F being Functor of [:A,B:],C
for a being Object of A holds (export F) . a = F ?- a
proofend;

theorem Th21: :: ISOCAT_2:21
for A, B, C being Category
for F being Functor of [:A,B:],C
for a being Object of A holds (export F) . a is Functor of B,C
proofend;

theorem Th22: :: ISOCAT_2:22
for A, B, C being Category
for F1, F2 being Functor of [:A,B:],C st export F1 = export F2 holds
F1 = F2
proofend;

theorem Th23: :: ISOCAT_2:23
for A, B, C being Category
for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds
( export F1 is_naturally_transformable_to export F2 &amp; ex G being natural_transformation of export F1, export F2 st
for s being Function of [: the carrier of A, the carrier of B:], the carrier' of C st t = s holds
for a being Object of A holds G . a = [[((export F1) . a),((export F2) . a)],((curry s) . a)] )
proofend;

definition
let A, B, C be Category;
let F1, F2 be Functor of [:A,B:],C;
assume B1: F1 is_naturally_transformable_to F2 ;
let t be natural_transformation of F1,F2;
func export t ->  natural_transformation of export F1, export F2 means :Def5: :: ISOCAT_2:def 5
 for s being Function of [: the carrier of A, the carrier of B:], the carrier' of C st t = s holds
for a being Object of A holds it . a = [[((export F1) . a),((export F2) . a)],((curry s) . a)];
existence
 ex b1 being natural_transformation of export F1, export F2 st
for s being Function of [: the carrier of A, the carrier of B:], the carrier' of C st t = s holds
for a being Object of A holds b1 . a = [[((export F1) . a),((export F2) . a)],((curry s) . a)] by B1, Th23;
uniqueness
 for b1, b2 being natural_transformation of export F1, export F2 st ( for s being Function of [: the carrier of A, the carrier of B:], the carrier' of C st t = s holds
for a being Object of A holds b1 . a = [[((export F1) . a),((export F2) . a)],((curry s) . a)] ) &amp; ( for s being Function of [: the carrier of A, the carrier of B:], the carrier' of C st t = s holds
for a being Object of A holds b2 . a = [[((export F1) . a),((export F2) . a)],((curry s) . a)] ) holds
b1 = b2
proofend;
end;

:: deftheorem Def5 defines export ISOCAT_2:def_5_:_
 for A, B, C being Category
for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2
for b7 being natural_transformation of export F1, export F2 holds
( b7 = export t iff for s being Function of [: the carrier of A, the carrier of B:], the carrier' of C st t = s holds
for a being Object of A holds b7 . a = [[((export F1) . a),((export F2) . a)],((curry s) . a)] );

theorem Th24: :: ISOCAT_2:24
for A, B, C being Category
for F being Functor of [:A,B:],C holds id (export F) = export (id F)
proofend;

theorem Th25: :: ISOCAT_2:25
for A, B, C being Category
for F1, F2, F3 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 &amp; F2 is_naturally_transformable_to F3 holds
for t1 being natural_transformation of F1,F2
for t2 being natural_transformation of F2,F3 holds export (t2 `*` t1) = (export t2) `*` (export t1)
proofend;

theorem Th26: :: ISOCAT_2:26
for A, B, C being Category
for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t1, t2 being natural_transformation of F1,F2 st export t1 = export t2 holds
t1 = t2
proofend;

theorem Th27: :: ISOCAT_2:27
for A, B, C being Category
for G being Functor of A, Functors (B,C) ex F being Functor of [:A,B:],C st G = export F
proofend;

theorem Th28: :: ISOCAT_2:28
for A, B, C being Category
for F1, F2 being Functor of [:A,B:],C st export F1 is_naturally_transformable_to export F2 holds
for t being natural_transformation of export F1, export F2 holds
( F1 is_naturally_transformable_to F2 &amp; ex u being natural_transformation of F1,F2 st t = export u )
proofend;

definition
let A, B, C be Category;
func export (A,B,C) ->  Functor of Functors ([:A,B:],C), Functors (A,(Functors (B,C))) means :Def6: :: ISOCAT_2:def 6
 for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds it . [[F1,F2],t] = [[(export F1),(export F2)],(export t)];
existence
 ex b1 being Functor of Functors ([:A,B:],C), Functors (A,(Functors (B,C))) st
for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b1 . [[F1,F2],t] = [[(export F1),(export F2)],(export t)]
proofend;
uniqueness
 for b1, b2 being Functor of Functors ([:A,B:],C), Functors (A,(Functors (B,C))) st ( for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b1 . [[F1,F2],t] = [[(export F1),(export F2)],(export t)] ) &amp; ( for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b2 . [[F1,F2],t] = [[(export F1),(export F2)],(export t)] ) holds
b1 = b2
proofend;
end;

:: deftheorem Def6 defines export ISOCAT_2:def_6_:_
 for A, B, C being Category
for b4 being Functor of Functors ([:A,B:],C), Functors (A,(Functors (B,C))) holds
( b4 = export (A,B,C) iff for F1, F2 being Functor of [:A,B:],C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b4 . [[F1,F2],t] = [[(export F1),(export F2)],(export t)] );

registration
let A, B, C be Category;
cluster export (A,B,C) ->  isomorphic ;
coherence
 export (A,B,C) is isomorphic
proofend;
end;

theorem :: ISOCAT_2:29
for A, B, C being Category holds Functors ([:A,B:],C) ~= Functors (A,(Functors (B,C)))
proofend;

begin

theorem Th30: :: ISOCAT_2:30
for A, B, C being Category
for F1, F2 being Functor of A,B
for G being Functor of B,C st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds G * t = G * t
proofend;

definition
let A, B, C be Category;
let F be Functor of A,B;
let G be Functor of A,C;
:: original:<:
redefine func<:F,G:> ->  Functor of A,[:B,C:];
coherence
 <:F,G:> is Functor of A,[:B,C:]
proofend;
end;

definition
let A, B, C be Category;
let F be Functor of A,[:B,C:];
func Pr1 F ->  Functor of A,B equals :: ISOCAT_2:def 7
(pr1 (B,C)) * F;
correctness
coherence
(pr1 (B,C)) * F is Functor of A,B;
;
func Pr2 F ->  Functor of A,C equals :: ISOCAT_2:def 8
(pr2 (B,C)) * F;
correctness
coherence
(pr2 (B,C)) * F is Functor of A,C;
;
end;

:: deftheorem  defines Pr1 ISOCAT_2:def_7_:_
 for A, B, C being Category
for F being Functor of A,[:B,C:] holds Pr1 F = (pr1 (B,C)) * F;

:: deftheorem  defines Pr2 ISOCAT_2:def_8_:_
 for A, B, C being Category
for F being Functor of A,[:B,C:] holds Pr2 F = (pr2 (B,C)) * F;

theorem Th31: :: ISOCAT_2:31
for A, B, C being Category
for F being Functor of A,B
for G being Functor of A,C holds
( Pr1 <:F,G:> = F &amp; Pr2 <:F,G:> = G )
proofend;

theorem Th32: :: ISOCAT_2:32
for A, B, C being Category
for F, G being Functor of A,[:B,C:] st Pr1 F = Pr1 G &amp; Pr2 F = Pr2 G holds
F = G
proofend;

definition
let A, B, C be Category;
let F1, F2 be Functor of A,[:B,C:];
let t be natural_transformation of F1,F2;
func Pr1 t ->  natural_transformation of Pr1 F1, Pr1 F2 equals :: ISOCAT_2:def 9
(pr1 (B,C)) * t;
coherence
 (pr1 (B,C)) * t is natural_transformation of Pr1 F1, Pr1 F2 ;
func Pr2 t ->  natural_transformation of Pr2 F1, Pr2 F2 equals :: ISOCAT_2:def 10
(pr2 (B,C)) * t;
coherence
 (pr2 (B,C)) * t is natural_transformation of Pr2 F1, Pr2 F2 ;
end;

:: deftheorem  defines Pr1 ISOCAT_2:def_9_:_
 for A, B, C being Category
for F1, F2 being Functor of A,[:B,C:]
for t being natural_transformation of F1,F2 holds Pr1 t = (pr1 (B,C)) * t;

:: deftheorem  defines Pr2 ISOCAT_2:def_10_:_
 for A, B, C being Category
for F1, F2 being Functor of A,[:B,C:]
for t being natural_transformation of F1,F2 holds Pr2 t = (pr2 (B,C)) * t;

theorem Th33: :: ISOCAT_2:33
for A, B, C being Category
for F1, F2, G1, G2 being Functor of A,[:B,C:] st F1 is_naturally_transformable_to F2 &amp; G1 is_naturally_transformable_to G2 holds
for s being natural_transformation of F1,F2
for t being natural_transformation of G1,G2 st Pr1 s = Pr1 t &amp; Pr2 s = Pr2 t holds
s = t
proofend;

Lm2:  for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_transformable_to G1 &amp; F2 is_transformable_to G2 holds
for t1 being transformation of F1,G1
for t2 being transformation of F2,G2
for a being Object of A holds <:t1,t2:> . a = [(t1 . a),(t2 . a)]
proofend;

theorem Th34: :: ISOCAT_2:34
for A, B, C being Category
for F being Functor of A,B
for G being Functor of A,C
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds <:F,G:> . f = [(F . f),(G . f)] by FUNCT_3:59;

theorem Th35: :: ISOCAT_2:35
for A, B, C being Category
for F being Functor of A,B
for G being Functor of A,C
for a being Object of A holds <:F,G:> . a = [(F . a),(G . a)]
proofend;

Lm3:  for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_transformable_to G1 &amp; F2 is_transformable_to G2 holds
for t1 being transformation of F1,G1
for t2 being transformation of F2,G2
for a being Object of A holds <:t1,t2:> . a in Hom ((<:F1,F2:> . a),(<:G1,G2:> . a))
proofend;

theorem Th36: :: ISOCAT_2:36
for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_transformable_to G1 &amp; F2 is_transformable_to G2 holds
<:F1,F2:> is_transformable_to <:G1,G2:>
proofend;

definition
let A, B, C be Category;
let F1, G1 be Functor of A,B;
let F2, G2 be Functor of A,C;
assume A1: ( F1 is_transformable_to G1 &amp; F2 is_transformable_to G2 ) ;
let t1 be transformation of F1,G1;
let t2 be transformation of F2,G2;
func<:t1,t2:> ->  transformation of <:F1,F2:>,<:G1,G2:> equals :Def11: :: ISOCAT_2:def 11
<:t1,t2:>;
coherence
 <:t1,t2:> is transformation of <:F1,F2:>,<:G1,G2:>
proofend;
end;

:: deftheorem Def11 defines <: ISOCAT_2:def_11_:_
 for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_transformable_to G1 &amp; F2 is_transformable_to G2 holds
for t1 being transformation of F1,G1
for t2 being transformation of F2,G2 holds <:t1,t2:> = <:t1,t2:>;

theorem Th37: :: ISOCAT_2:37
for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_transformable_to G1 &amp; F2 is_transformable_to G2 holds
for t1 being transformation of F1,G1
for t2 being transformation of F2,G2
for a being Object of A holds <:t1,t2:> . a = [(t1 . a),(t2 . a)]
proofend;

Lm4:  for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_naturally_transformable_to G1 &amp; F2 is_naturally_transformable_to G2 holds
for t1 being natural_transformation of F1,G1
for t2 being natural_transformation of F2,G2
for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (<:t1,t2:> . b) * (<:F1,F2:> /. f) = (<:G1,G2:> /. f) * (<:t1,t2:> . a)
proofend;

theorem Th38: :: ISOCAT_2:38
for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_naturally_transformable_to G1 &amp; F2 is_naturally_transformable_to G2 holds
<:F1,F2:> is_naturally_transformable_to <:G1,G2:>
proofend;

definition
let A, B, C be Category;
let F1, G1 be Functor of A,B;
let F2, G2 be Functor of A,C;
assume A1: ( F1 is_naturally_transformable_to G1 &amp; F2 is_naturally_transformable_to G2 ) ;
let t1 be natural_transformation of F1,G1;
let t2 be natural_transformation of F2,G2;
func<:t1,t2:> ->  natural_transformation of <:F1,F2:>,<:G1,G2:> equals :Def12: :: ISOCAT_2:def 12
<:t1,t2:>;
coherence
 <:t1,t2:> is natural_transformation of <:F1,F2:>,<:G1,G2:>
proofend;
end;

:: deftheorem Def12 defines <: ISOCAT_2:def_12_:_
 for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_naturally_transformable_to G1 &amp; F2 is_naturally_transformable_to G2 holds
for t1 being natural_transformation of F1,G1
for t2 being natural_transformation of F2,G2 holds <:t1,t2:> = <:t1,t2:>;

theorem Th39: :: ISOCAT_2:39
for A, B, C being Category
for F1, G1 being Functor of A,B
for F2, G2 being Functor of A,C st F1 is_naturally_transformable_to G1 &amp; F2 is_naturally_transformable_to G2 holds
for t1 being natural_transformation of F1,G1
for t2 being natural_transformation of F2,G2 holds
( Pr1 <:t1,t2:> = t1 &amp; Pr2 <:t1,t2:> = t2 )
proofend;

definition
let A, B, C be Category;
func distribute (A,B,C) ->  Functor of Functors (A,[:B,C:]),[:(Functors (A,B)),(Functors (A,C)):] means :Def13: :: ISOCAT_2:def 13
 for F1, F2 being Functor of A,[:B,C:] st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds it . [[F1,F2],t] = [[[(Pr1 F1),(Pr1 F2)],(Pr1 t)],[[(Pr2 F1),(Pr2 F2)],(Pr2 t)]];
existence
 ex b1 being Functor of Functors (A,[:B,C:]),[:(Functors (A,B)),(Functors (A,C)):] st
for F1, F2 being Functor of A,[:B,C:] st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b1 . [[F1,F2],t] = [[[(Pr1 F1),(Pr1 F2)],(Pr1 t)],[[(Pr2 F1),(Pr2 F2)],(Pr2 t)]]
proofend;
uniqueness
 for b1, b2 being Functor of Functors (A,[:B,C:]),[:(Functors (A,B)),(Functors (A,C)):] st ( for F1, F2 being Functor of A,[:B,C:] st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b1 . [[F1,F2],t] = [[[(Pr1 F1),(Pr1 F2)],(Pr1 t)],[[(Pr2 F1),(Pr2 F2)],(Pr2 t)]] ) &amp; ( for F1, F2 being Functor of A,[:B,C:] st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b2 . [[F1,F2],t] = [[[(Pr1 F1),(Pr1 F2)],(Pr1 t)],[[(Pr2 F1),(Pr2 F2)],(Pr2 t)]] ) holds
b1 = b2
proofend;
end;

:: deftheorem Def13 defines distribute ISOCAT_2:def_13_:_
 for A, B, C being Category
for b4 being Functor of Functors (A,[:B,C:]),[:(Functors (A,B)),(Functors (A,C)):] holds
( b4 = distribute (A,B,C) iff for F1, F2 being Functor of A,[:B,C:] st F1 is_naturally_transformable_to F2 holds
for t being natural_transformation of F1,F2 holds b4 . [[F1,F2],t] = [[[(Pr1 F1),(Pr1 F2)],(Pr1 t)],[[(Pr2 F1),(Pr2 F2)],(Pr2 t)]] );

registration
let A, B, C be Category;
cluster distribute (A,B,C) ->  isomorphic ;
coherence
 distribute (A,B,C) is isomorphic
proofend;
end;

theorem :: ISOCAT_2:40
for A, B, C being Category holds Functors (A,[:B,C:]) ~= [:(Functors (A,B)),(Functors (A,C)):]
proofend;