reserve C for CategoryStr;
reserve f,f1,f2,f3 for morphism of C;
reserve g1,g2 for morphism of C opp;

theorem Th22:
  for C being non empty with_identities CategoryStr, f being morphism of C
  holds f is identity iff f is Object of C
  proof
    let C be non empty with_identities CategoryStr;
    let f be morphism of C;
    hereby
      assume f is identity;
      then f in {u where u is morphism of C: u is identity & u in Mor(C)};
      hence f is Object of C;
    end;
    assume f is Object of C;
    then f in {u where u is morphism of C: u is identity & u in Mor(C)};
    then consider u be morphism of C such that
A1: f = u & u is identity & u in Mor(C);
    thus f is identity by A1;
  end;
