Theory of categories is all natural and abstract nonsense. Or is it? What would be the most unexpected and/or the least natural theorem of the theory of category? (It does not really have to be THE).
The Dirichlet pigeonhole principle, even if it belonged to the theory of categories (it does not), it would not provide the answer to my question. Indeed, its applications are striking. But the principle itself is obvious and natural.
In topos theory, it is not at all obvious that an elementary topos has finite colimits: there is nothing in the usual definition which would suggest this. And the standard proof, which passes through monadicity theorems and Beck-Chevalley conditions, winds up looking highly technical to most people, and (for most people) doesn't shed much intuitive light on why the result is true.
It's possible to prove this result by other means. There is some information on constructing coproducts in a topos in a hands-on way here, but as one can see it takes a while to set out. One can do this for coequalizers as well, but I don't know where this has been written down (yet).
In a similar vein to Todd Trimble's remark on elementary toposes, let's recall the following fact:
Theorem. An accessible category is complete if and only if it is cocomplete.
The quickest proof I know of uses the accessible adjoint functor theorem. The curious thing is this: from a theoretical perspective, the cocompleteness hypothesis is more useful; yet in practice it is the completeness hypothesis that is easiest to verify. For instance, it is easy to see that many categories of essentially algebraic structures (e.g. groups, rings, categories) are accessible and complete, but the construction of colimits in these categories is often non-obvious.
The only instance of the "if" direction I am aware of is a corollary of Giraud's theorem: any category satisfying Giraud's axioms is necessarily complete. This is obvious if we assume Giraud's theorem, because any such category is a topos of sheaves on a Grothendieck site, hence, a reflective subcategory of a complete category.
If Yoneda's Lemma counts as "unexpected", then this would be my favorite (but you might argue that it is too elementary to be unexpected).
It has numerous applications, and I particularly like the framework it provides to define affine varieties as representable functors from the category of commutative $k$-algebras to the category of sets, and similarly to define linear algebraic groups as representable functors from the category of commutative $k$-algebras to the category of groups (more precisely those functors represented by a finitely generated $k$-algebra).
In particular, Yoneda's Lemma gives, without any additional effort, the duality between the category of linear algebraic $k$-groups and the category of finitely generated commutative Hopf $k$-algebras.
