* Size issues. All the problems in set theory are back with a vengeance. Exploring size issues makes it clear that Russell's Paradox is not a fluke.
* Topoi. A topos is a generalization of the past few millennia of maths, and their existence shows that maths is a multiverse rather than a single collection of rules. Some topoi witness refutations of LEM, making them valuable counterexamples.
* Yoneda's lemma. The core of pragmatic philosophy is that objects are not separable from their effects in the world. Yoneda's result takes that to its logical conclusion: An object is equivalent to all of its effects under transformation.
* Equivalence of categories. Rel is equivalent to its dual (Rel is daggered) and so is Hilb; the category Set* of sets with an extra point is equivalent to the category Pfn of sets and partial functions. These show us that parts of maths are self-symmetric beyond group theory.
In addition, why shouldn't adjunctions be covered at some point? Adjunctions can't be studied without category theory (look up "Galois connections" for a history) and they're hard to avoid once we're talking about enough categories of note. I bet that the adjointness of syntax and semantics is very relevant to philosophy of language.