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 Th56:
  for T being Function of the carrier' of C,the carrier' of D st
   (for c being Object of C ex d being Object of D st T.(id c) = id d) &
   (for f being Morphism of C
     holds T.(id dom f) = id dom (T.f) &
           T.(id cod f) = id 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 such that
A1: for c being Object of C ex d being Object of D st T.(id c) = id d and
A2: for f being Morphism of C holds T.(id dom f) = id dom (T.f) & T.(id
  cod f) = id 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);
  thus for c being Element of C ex d being Element of D st T.id c = id d
                 by A1;
  thus for f being Element of the carrier' of C
   holds T.(id dom f) = id dom(T.f) &
     T.(id cod f) = id cod(T.f) by A2;
  let f,g be Element of the carrier' of C;
  assume [g,f] in dom(the Comp of C);
  then
A4: dom g = cod f by Def4;
  thus thesis by A3,A4;
end;
