I like this objection, because it illustrates why category theory (or the category-theoretical way of thinking) is valuable even in elementary discussions. “Is

I like this objection, because it illustrates why category theory (or the category-theoretical way of thinking) is valuable even in elementary discussions. “Isomorphism” is a great example of a word that is not well-defined out of context. An isomorphism of rings is not the same as an isomorphism of groups, for example. In category theory, an isomorphism is a morphism that has an inverse. Category theorists sometimes use the punning device of treating “iso” as an adjective, so that an “isomorphism” is an “iso morphism”. But before you can talk about morphisms, you need to know which category you‘re talking about. An isomorphism in the category of sets is indeed precisely the same thing as a bijection, so in that sense bijections are necessarily isomorphisms. But the word is traditionally used in the context of categories of algebras (e.g. groups, rings, fields, etc.): the morphisms of those categories are the homomorphisms (i.e. functions that preserve the algebraic structure), and so of course the iso morphisms in those categories are also homomorphisms.