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 Th60:
  for T being Function of the carrier' of C,the carrier' of D for
F being Function of the carrier of C, the carrier of D st ( for c being Object
of C holds T.(id c) = id(F.c) ) & ( for f being Morphism of C holds F.(dom f) =
dom (T.f) & F.(cod f) = cod (T.f) ) & ( for f,g being Morphism of C st dom g =
  cod f holds T.(g(*)f) = (T.g)(*)(T.f)) holds T is Functor of C,D
proof
  let T be Function of the carrier' of C,the carrier' of D;
  let F be Function of the carrier of C, the carrier of D;
  assume that
A1: for c being Object of C holds T.(id c) = id(F.c) and
A2: for f being Morphism of C holds F.(dom f) = dom (T.f) & F.(cod f) =
  cod (T.f) and
A3: for f,g being Morphism of C st dom g = cod f
             holds T.(g(*)f) = (T.g)(*)( T.f);
A4: for c being Object of C ex d being Object of D st T.(id c) = id d
  proof
    let c be Object of C;
    take F.c;
    thus thesis by A1;
  end;
  for f being Morphism of C holds T.(id dom f) = id dom (T.f) & T.(id cod
  f) = id cod (T.f)
  proof
    let f be Morphism of C;
    thus T.(id dom f) = id (F.(dom f)) by A1
      .= id dom (T.f) by A2;
    thus T.(id cod f) = id (F.(cod f)) by A1
      .= id cod (T.f) by A2;
  end;
  hence thesis by A3,A4,Th56;
end;
