reserve C for CategoryStr;
reserve f,f1,f2,f3 for morphism of C;
reserve g1,g2 for morphism of C opp;
reserve C,D,E for with_identities CategoryStr;
reserve F for Functor of C,D;
reserve G for Functor of D,E;
reserve f for morphism of C;

theorem Th32:
  for C,D being non empty composable with_identities CategoryStr,
      F being covariant Functor of C,D,
      f being morphism of C
  holds F.(dom f) = dom(F.f) & F.(cod f) = cod(F.f)
  proof
    let C,D be non empty composable with_identities CategoryStr;
    let F be covariant Functor of C,D;
    let f be morphism of C;
A1: F is multiplicative by Def25;
    consider d be morphism of C such that
A2: dom f = d & f |> d & d is identity by Def18;
    F.(dom f) in Ob(D);
    then reconsider d1 = F.(dom f) as morphism of D;
A3: F.f |> F.d by A2,A1;
A4: F.d = F.(dom f) by A2,Def21;
    d1 is identity by Th22;
    hence F.(dom f) = dom(F.f) by A3,A4,Def18;
    consider c be morphism of C such that
A5: cod f = c & c |> f & c is identity by Def19;
    F.(cod f) in Ob(D);
    then reconsider c1 = F.(cod f) as morphism of D;
A6: F.c |> F.f by A5,A1;
A7: F.c = F.(cod f) by A5,Def21;
    c1 is identity by Th22;
    hence F.(cod f) = cod(F.f) by A6,A7,Def19;
  end;
