Theorem 1.2.10 (Equivalence).label Let $F:\frak{C}\to \frak{D}$ be a covariant (resp. contravariant) functor. There exists a functor $G:\frak{D}\to \frak{C}$ of the same variance such that $(F,G)$ is an equivalence (resp. antiequivalence) of categories if and only if:

• $F$ is full i.e. surjective on morphisms

• $F$ is faithful i.e. injective on morphisms

• $F$ is essentially surjective i.e. surjective on objects up to isomorphism