reserve C for CatStr;
reserve f,g for Morphism of C;
reserve C for non void non empty CatStr,
  f,g for Morphism of C,
  a,b,c,d for Object of C;
reserve o,m for set;
reserve B,C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,f1,f2,g,g1,g2 for Morphism of C;
reserve f,f1,f2 for Morphism of a,b;
reserve f9 for Morphism of b,a;
reserve g for Morphism of b,c;
reserve h,h1,h2 for Morphism of c,d;

theorem Th83:
  for T being Functor of C,D for c,c9 being Object of C st Hom(c,
  c9) <> {} for f being Morphism of c,c9 holds hom(T,c,c9).f = T.f
proof
  let T be Functor of C,D;
  let c,c9 be Object of C;
  assume
A1: Hom(c,c9) <> {};
  let f be Morphism of c,c9;
  f in Hom(c,c9) by A1,Def3;
  hence thesis by FUNCT_1:49;
end;
