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 Th70:
  for T being (Functor of B,C),S being (Functor of C,D),b being
  Object of B holds (Obj (S*T)).b = (Obj S).((Obj T).b)
proof
  let T be (Functor of B,C),S be (Functor of C,D),b be Object of B;
A1: (S*T).(id b) = S.(T.(id b)) by FUNCT_2:15;
  consider d being Object of D such that
A2: (S*T).(id b) = id d by Def19;
  consider c being Object of C such that
A3: T.(id b) = id c by Def19;
  thus (Obj (S*T)).b = d by A2,Th62
    .= (Obj S).c by A2,A3,A1,Th62
    .= (Obj S).((Obj T).b) by A3,Th62;
end;
