Say I'm just a humble mathematician, not having already drunk the category theory kool-aid. I want to convince myself that the definition of category is "sensible" or "good" or whatever. I could try to look to the fact that it appears to characterize how sets and functions behave. The reason, I tell myself, that associativity is required, is because composition of functions between sets really is associative. The reason that we require identities is that, for any set, there does inevitably exist an identity function on it.
Now, the Yoneda lemma is a crucial formal reflection of this intuition: every category can in fact be seen as a concrete category made up of sets and functions.
Question 1: is there a kind of converse to the Yoneda lemma that feels like a model-theoretic completeness theorem? Does the "theory of a category" (i.e. "there are some objects and some arrows and identities and composites and the composites are associative") capture everything that's true, in an appropriate sense, of sets and functions? Or am I thinking of this backwards and the Yoneda lemma actually is in the completeness direction? I'm not sure.
Question 2: What do I replace "sets and functions" with to get bicategories? What's the simplest ("free-est") concrete mathematical data structure that I want to use in place of function, that has the property that it's associative up to isomorphism, but not on the nose?
The first things that come to mind for Q2 --- the examples of non-strict bicategories most familiar to me --- are topological-spaces/continuous functions/homotopies and sets/spans of sets/maps-of-spans. But the homotopy bicategory seems needlessly complicated --- surely there's some simpler, more plain-set-theoretic example that works? And spans aren't "free" enough, I think. It is a true fact that for any span there's a converse span, but this dualization isn't available in a general bicategory.
Oh, wait, um, the homotopy bicategory also admits time-reversal of homotopies, doesn't it. So scratch that, too.
I tried reading about the Yoneda lemma for bicategories hoping for clarification, but instead was met with the usual incomprehensible zoo of notions of subtly different weakened thingies and weakened equivalences. And I wish I had more concrete litmus tests and diagnostic tools for understanding such things, which is why I ask these questions in the first place.
Hm, the more I think about it, the more I think Q1 is answered by "yes, the Yoneda lemma is a kind of completeness theorem". If there were some property that "all models satisfied" (i.e. all collections of sets and functions between them satisfied) but was not captured by the axiomatization of categories that we know, then converting some category that failed to have that property to a concrete category should fail.
I'm still curious about Q2 though.
Wait, no, I still hunger for some more satisfying answer to Q1 after all. I could just as truthfully say "every category can be faithfully embedded into the category of sets and relations" rather than sets and functions, but the axiomatic theory-of-a-category doesn't let me form the converse of an arrow, which is nonetheless a semantically legitimate thing to do with relations.