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 Th59:
  for T being Functor of C,D for f,g being Morphism of C st dom g
  = cod f holds dom(T.g) = cod(T.f) & T.(g(*)f) = (T.g)(*)(T.f)
proof
  let T be Functor of C,D;
  let f,g be Morphism of C;
  assume
A1: dom g = cod f;
  then
A2: (the Comp of C).(g,f) = g(*)f & [g,f] in dom(the Comp of C)
     by Def4,Th14;
 id dom (T.g) = T.(id cod f) by A1,Def19
    .= id cod (T.f) by Def19;
  hence dom (T.g) = cod (T.f) by Th54;
  thus thesis by A2,Def19;
end;
