
theorem Th44:
  for A being category, a, b being Object of A st <^a,b^> <> {}
  for f being Morphism of a,b ex F being Function of
  (Concretized A)-carrier_of a, (Concretized A)-carrier_of b
  st F in (the Arrows of Concretized A).(a,b) &
  for c being Object of A, g being Morphism of c,a st <^c,a^> <> {}
  holds F.[g, [c,a]] = [f*g, [c,b]]
proof
  let A be category, a, b be Object of A such that
A1: <^a,b^> <> {};
  set B = Concretized A;
  let f be Morphism of a,b;
  defpred P[object,object] means
  ex o being Object of A, g being Morphism of o, a st
  <^o,a^> <> {} & $1 = [g,[o,a]] & $2 = [f*g, [o,b]];
A2: for x being object st x in B-carrier_of a
  ex y being object st y in B-carrier_of b & P[x, y]
  proof
    let x be object;
    assume x in B-carrier_of a;
    then consider o being Object of A, g being Morphism of o,a such that
A3: <^o,a^> <> {} and
A4: x = [g,[o,a]] by Th43;
    take [f*g, [o,b]];
    <^o,b^> <> {} by A1,A3,ALTCAT_1:def 2;
    hence thesis by A3,A4,Th43;
  end;
  consider F being Function such that
A5: dom F = B-carrier_of a & rng F c= B-carrier_of b and
A6: for x being object st x in B-carrier_of a holds P[x, F.x]
  from FUNCT_1:sch 6(A2);
A7: F in Funcs(B-carrier_of a, B-carrier_of b) by A5,FUNCT_2:def 2;
  then reconsider F as Function of B-carrier_of a, B-carrier_of b by FUNCT_2:66
;
  take F;
  ex fa,fb being Object of A, g being Morphism of fa, fb
  st fa = a & fb = b & <^fa, fb^> <> {} &
  for o being Object of A st <^o, fa^> <> {}
  for h being Morphism of o,fa holds F.[h,[o,fa]] = [g*h,[o,fb]]
  proof
    take fa = a, fb = b;
    reconsider g = f as Morphism of fa,fb;
    take g;
    thus fa = a & fb = b & <^fa, fb^> <> {} by A1;
    let o be Object of A such that
A8: <^o, fa^> <> {};
    let h be Morphism of o,fa;
    [h,[o,fa]] in B-carrier_of fa by A8,Th43;
    then consider c being Object of A, k being Morphism of c, fa such that
    <^c,fa^> <> {} and
A9: [h,[o,fa]] = [k,[c,fa]] and
A10: F.[h,[o,fa]] = [g*k, [c,fb]] by A6;
    [o,fa] = [c,fa] by A9,XTUPLE_0:1;
    then o = c by XTUPLE_0:1;
    hence thesis by A9,A10,XTUPLE_0:1;
  end;
  hence F in (the Arrows of B).(a,b) by A7,Def12;
  let c be Object of A, g be Morphism of c,a;
  assume <^c,a^> <> {};
  then [g, [c,a]] in B-carrier_of a by Th43;
  then consider o being Object of A, h being Morphism of o, a such that
  <^o,a^> <> {} and
A11: [g,[c,a]] = [h,[o,a]] and
A12: F.[g,[c,a]] = [f*h, [o,b]] by A6;
  [c,a] = [o,a] by A11,XTUPLE_0:1;
  then c = o by XTUPLE_0:1;
  hence thesis by A11,A12,XTUPLE_0:1;
end;
