reserve C for CatStr;
reserve f,g for Morphism of C;
reserve C for non void non empty CatStr,
  f,g for Morphism of C,
  a,b,c,d for Object of C;
reserve o,m for set;
reserve B,C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,f1,f2,g,g1,g2 for Morphism of C;
reserve f,f1,f2 for Morphism of a,b;
reserve f9 for Morphism of b,a;
reserve g for Morphism of b,c;
reserve h,h1,h2 for Morphism of c,d;

theorem
 the carrier' of C = union the set of all Hom(a,b)
proof
  set A = the set of all Hom(a,b), M = union A;
 thus the carrier' of C c= M
  proof let e be object;
   assume e in the carrier' of C;
    then reconsider e as Morphism of C;
A1:  e in Hom(dom e,cod e);
   Hom(dom e,cod e) in A;
   hence thesis by A1,TARSKI:def 4;
  end;
 let e be object;
  assume e in M;
   then consider X being set such that
A2:  e in X and
A3:  X in A by TARSKI:def 4;
  ex a,b st X = Hom(a,b) by A3;
 hence thesis by A2;
end;
