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 Th76:
  for T being Functor of C,D for c,c9 being Object of C for f
  being set st f in Hom(c,c9) holds T.f in Hom(T.c,T.c9)
proof
  let T be Functor of C,D;
  let c,c9 be Object of C;
  let f be set;
  assume
A1: f in Hom(c,c9);
  then reconsider f9 = f as Morphism of c,c9 by Def3;
  cod f9 = c9 by A1,Th1;
  then
A2: T.c9 = cod(T.f9) by Th64;
  dom f9 = c by A1,Th1;
  then T.c = dom (T.f9) by Th64;
  hence thesis by A2;
end;
