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 Th51:
  for C1,C2 being Category st the carrier' of C1 = the carrier' of C2
  & the Comp of C1 = the Comp of C2 holds C1 ~= C2
  proof
    let C1,C2 be Category;
    assume the carrier' of C1 = the carrier' of C2 &
      the Comp of C1 = the Comp of C2;
    then alter(C1) = alter(C2);
    hence C1 ~= C2 by Th50;
  end;
