The Stacks project

Definition 4.2.17. An equivalence of categories $F : \mathcal{A} \to \mathcal{B}$ is a functor such that there exists a functor $G : \mathcal{B} \to \mathcal{A}$ such that the compositions $F \circ G$ and $G \circ F$ are isomorphic to the identity functors $\text{id}_\mathcal {B}$, respectively $\text{id}_\mathcal {A}$. In this case we say that $G$ is a quasi-inverse to $F$.