An equivalence of categories consists of functors $F \colon \mathsf{C} \leftrightarrows \mathsf{D} \colon G$ together with natural isomorphisms $\eta \colon 1\mathsf{C} \cong GF$, $\epsilon \colon FG \cong 1\mathsf{D}$. Categories $\mathsf{C}$ and $\mathsf{D}$ are equivalent, written $\mathsf{C} \simeq\mathsf{D}$, if there exists an equivalence between them.