The automorphism h−1০g of E' extended by IdX' forms an automorphism of the model (E'∪E'') of T''. As j is invariant, it stays equal to its image j০(h−1০g)−1 by this automorphism. Therefore h−1০g = IdE' , i.e. g=h.What we can see here strictly preserved by constructions, is the class of isomorphisms as an abstract category. In particular, constructions leave unchanged the automorphism groups seen as abstract groups, providing more types on which they act.