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 Th45:
  for C being non empty category, f1 being morphism of C,
      a1 being Morphism of Alter(C) st a1 = f1
  holds dom f1 = dom a1 & cod f1 = cod a1
  proof
    let C be non empty category;
    let f1 be morphism of C;
    let a1 be Morphism of Alter(C);
    assume
A1: a1 = f1;
    thus dom f1 = (the Source of Alter(C)).a1 by A1,Def30
    .= dom a1 by GRAPH_1:def 3;
    thus cod f1 = (the Target of Alter(C)).a1 by A1,Def31
    .= cod a1 by GRAPH_1:def 4;
  end;
