Fundamental Theorem of Category Theory appropriate for undergraduates? I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics build up to a “Fundamental Theorem” that is a somewhat deep result in the subject, is accessible to undergraduate level students in the time available, and serves as a kind of “capstone” which brings together the techniques and theorems studied in the course. For example, an introduction to abstract algebra and group theory may have the fundamental theorem of finite abelian groups, and an introduction to analysis may have the fundamental theorem of calculus. Is there an analogous result in Category theory that meets these criteria? Does category theory have too many prerequisites for reaching such a result? What would be a good result to conclude a short introduction to category theory with?
 A: As mentioned above I believe the Yoneda lemma is a good candidate.  
It requires an understanding of categories, functors, exponentials, natural transformations and their interplay -- how to parametrize natural transformations between the contravariant representable functors of a category $\mathcal{C}$ in terms of arrows in $\mathcal{C}$ and vice-verse.
As mentioned by Todd Trimble above the proof is short and sweet, probably easier than most standard capstone proofs, but I agree that it is 'deep' -- it elucidates something fundamental about categories themselves rather than how category theory can be applied elsewhere.
It does miss out on machinery like limits/colimits which are mentioned in many adjoint functor theorems; perhaps that 'right adjoints preserve limits' could be an additional 'capstone' theorem in the opposite direction, slightly easier for an undergraduate class than the standard adjoint functor theorems.
A: The Special Adjoint Functor Theorem has already been recommended, and I agree with that suggestion. I also nominate the idea that "All concepts are Kan Extensions" as a capstone. Classically, you might finish the course with MacLane's coherence theorem, but I prefer to end with something that has lots of applications students would appreciate. 
Another example (that I might select, as a homotopy theorist) would be Giraud's theorem. 
A good resource is Emily Riehl's book Category Theory in Context, which is aimed at undergraduates and finishes with an Epilogue titled "Theorems in Category Theory".
