Equivalence of categories

Summary

An equivalence of categories is a functor $F \colon \mathcal{C} \to \mathcal{D}$ that has a functor $G \colon \mathcal{D} \to \mathcal{C}$ in the other direction. The composites $G \circ F$ and $F \circ G$ must be naturally isomorphic to the identity functors. It is the right notion of "sameness" for categories: it ignores how many copies of each object up to isomorphism a category has. A functor is an equivalence exactly when it is fully faithful and essentially surjective, by this theorem. Every category is equivalent to its skeleton.