Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
-
An equivalence of categories between $\mathcal{C}$ and $\mathcal{D}$ consists of a pair of functors
\begin{align*} F & \colon \mathcal{C}\to \mathcal{D},\\ G & \colon \mathcal{D}\to \mathcal{C} \end{align*}
together with natural isomorphisms
\begin{align*} \eta & \colon \text{id}_{\mathcal{C}} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}G\circ F,\\ \epsilon & \colon F\circ G \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\text{id}_{\mathcal{D}}. \end{align*} - An adjoint equivalence of categories between $\mathcal{C}$ and $\mathcal{D}$ is an equivalence $\webleft (F,G,\eta ,\epsilon \webright )$ between $\mathcal{C}$ and $\mathcal{D}$ which is also an adjunction.
