This isn't really a research-level question (sorry!), but I asked on math.se (link), and though the

questionwas upvoted a few times, I didn't get any answers. So since there may well be more category theorists hanging out here, let me try again!

In familiar introductory books on category theory, one of the very first examples of a category given is **Set**. And what category is that?

Typically no explanation is given at this stage. But of course which category we are dealing with depends on our set theory. For an NF-iste, the category of **NFsets** has very different properties from the usual category **Set** (for a start **NFsets** is not cartesian closed). But fair enough, in an intro book you aren't going to mention that in Ch. 1! No: the authors are, surely, intending to point to stuff that their beginning readers can be assumed to know about, and are saying, "Hey you in fact already know about some categories, for example ..." The charitable reading, then, is that authors are relying on their readers to think of **Set** as comprising the sets they already know and love from their standard intro set-theory course.

Which are pure sets of the cumulative hierarchy -- pure in that there are no urlements, no memberless entities in the universe of sets other than the empty sets. [If set theories with urelements are mentioned in passing in an intro course, it is usually only to be dismissed and forgotten about.]

OK, then: in the absence of special explicit signals to the contrary, it seems (doesn't it?) that we might reasonably take the category **Set** mentioned in the very early pages to be a category of pure sets of the usual hierarchy. What else?

But then what are we to make, a bit later in the book, of e.g. the usual presentation of the Yoneda embedding as $\mathcal{Y}\colon \mathscr{C} \to [\mathscr{C}^{op}, \mathbf{Set}]$. Putting it this way assumes that hom-collections $\mathscr{C}(A, B)$ for $A, B \in \mathscr{C}$ actually live in $\mathbf{Set}$. And since such a hom-collection is a set of $\mathscr{C}$-arrows, that assumes that the $\mathscr{C}$-arrows must live in the world of pure sets too. [We may want the relevant hom-collections to be set-sized in the Yoneda embedding case -- but being no bigger than set-sized is one thing, living in the universe of pure sets is something else!]

*But do we really want to assume that arrows [in the small-enough categories] are always pure sets?* Isn't category theory supposed to be a story about how different bits of the mathematical universe hang together which *doesn't* presuppose some over-arching, all-in, set-theoretic reductionism, and so in particular doesn't presuppose from day one that all morphisms are pure sets??

Now, the foundational sections you often meet early in category theory books worry away about questions of *size* (sets vs classes etc.). But the present worry is orthogonal to all that, and is in a way more basic. If we want to make no assumption that the denizens of different bits of the mathematical universe are all cut from the same cloth, we won't want to slip into assuming that sets of these denizens are all pure sets. So in particular, do we really want to assume that a collection of arrows (hom-set) must live in $\mathbf{Set}$ -- where that's the category mentioned back almost on p.1 of the book -- (as opposed, perhaps, to being fully faithfully mappable into that world?

I guess there must be good discussions of this sort of thing in the literature somewhere, and I'm no doubt showing my ignorance by asking where! But, please, any pointers would be most gratefully received.

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