To really drive the point home of the level of abstraction I like to think of a category of domino tiles. It kind of makes a few things apparent. Tiles would

To really drive the point home of the level of abstraction I like to think of a category of domino tiles. It kind of makes a few things apparent. Tiles would be the morphisms and objects would be the number spots on the tile ends. So the first insight: morphisms is the actaul object of study. The “objects” of a category is just a consequence of how the morphisms may, or may not, compose. My first hurdle when learning things was being distracted by trying to understand what objects where, some times they would be types, sometimes sets, and it was never really clear what their cardinality would be. As if the theory hid important details that you had to keeps track of implicitly. But it doesn’t, objects really are inconsequential, they are just there to be a place for morphisms to meet, nothing more. Just like the number of dots of the domino tiles. We could have used color instead, and the tiles would compose in exactly the same way. Second intuition, there’s no computation or anything else going on. Just as you say about the functions. The only thing of concern are the composability of those things called morphisms, be they functions or domino tiles. In the lecture series linked a few places in this thread, Bartosz makes a fantastic point of this when deriving some properties of surjective and bijective functions using only composability.