How do isomorphisms determine equivalence relations on the class of all groups?

Question

Let $\phi:G_{1}\to G_{2}$ and $\psi:G_{2} \to G_{3}$ be isomorphisms and assume that $\phi ^{-1}$ and $\psi \circ \phi$ are isomorphisms. Show that the isomorphism of groups determines an equivalence relation on the class of all groups.

Is it sufficient to just say that $\phi ^{-1}$ represents symmetry and $\psi \circ \phi$ represents transitivity? Then what about reflexivity? Is the identity function an isomorphism?

Answer 1

Yes, the $id:G \to G$ s.t $id(g)=g; \forall g \in G$ gives you the reflexivity.