An isomorphism of categories is a pair of functors
\begin{align*} F & \colon \mathcal{C}\to \mathcal{D},\\ G & \colon \mathcal{D}\to \mathcal{C} \end{align*}
such that we have
\begin{align*} G\circ F & = \text{id}_{\mathcal{C}},\\ F\circ G & = \text{id}_{\mathcal{D}}. \end{align*}
