If you want, it's even possible to fully commit to the isomorphism point of view by saying that you'll represent the structures themselves by their identity iso

If you want, it's even possible to fully commit to the isomorphism point of view by saying that you'll represent the structures themselves by their identity isomorphisms, but then if you "delete" the little tag telling you which particular endo was the identity (would a physicist say "up to phase"?), you might discover other interesting things...