reserve A,B,C for Category,
  F,F1 for Functor of A,B;
reserve o,m for set;
reserve t for natural_transformation of F,F1;

theorem Th10:
  for a1,a2 being Object of A, b1,b2 being Object of B st Hom([a1,
b1],[a2,b2]) <> {} for f being (Morphism of A), g being Morphism of B holds [f,
g] is Morphism of [a1,b1],[a2,b2] iff f is Morphism of a1,a2 & g is Morphism of
  b1,b2
proof
  let a1,a2 be Object of A, b1,b2 be Object of B;
  assume
A1: Hom([a1,b1],[a2,b2]) <> {};
  let f be Morphism of A;
  let g be Morphism of B;
  thus [f,g] is Morphism of [a1,b1],[a2,b2] implies f is Morphism of a1,a2 & g
  is Morphism of b1,b2
  proof
    assume [f,g] is Morphism of [a1,b1],[a2,b2];
    then
A2: [f,g] in Hom([a1,b1],[a2,b2]) by A1,CAT_1:def 5;
    Hom([a1,b1],[a2,b2]) = [:Hom(a1,a2),Hom(b1,b2):] by CAT_2:32;
    then f in Hom(a1,a2) & g in Hom(b1,b2) by A2,ZFMISC_1:87;
    hence thesis by CAT_1:def 5;
  end;
  Hom(a1,a2) <> {} & Hom(b1,b2) <> {} by A1,Th9;
  hence thesis by CAT_2:33;
end;
