From Milewski: "That’s because category theory — rather than dealing with particulars — deals with structure. It deals with the kind of structure that makes programs composable."

And he is right, because morphisms may or may not preserve structure. If you want to nitpick and create structure from the absurd function morphism - then yeah, so I think a discussion about this gets tedious. The more you look into the matter the more structure / data and morphisms merge and your point feels more like an invitation for the newbies to have a mental breakdown.