
theorem Th9:
  for C1,C2 being category, F being Functor of C1,C2 st
  F is isomorphism holds F is bijective
  proof
    let C1,C2 be category;
    let F be Functor of C1,C2;
    assume
A1: F is isomorphism;
    then
A2: F is covariant by CAT_7:def 19;
    consider G be Functor of C2,C1 such that
A3: G is covariant & G (*) F = id C1 & F (*) G = id C2 by A1,CAT_7:def 19;
A4: F*G = id C1 & G*F = id C2 by A2,A3,CAT_6:def 27;
    id C1 = id the carrier of C1 & id C2 = id the carrier of C2
    by STRUCT_0:def 4;
    then F is one-to-one & F is onto by A4,FUNCT_2:23;
    hence F is bijective;
  end;
