A morphism $f\colon A\to B$ of $\mathcal{C}$ is an isomorphism if there exists a morphism $\smash {f^{-1}\colon B\to A}$ of $\mathcal{C}$ such that
\begin{align*} f\circ f^{-1} & = \operatorname {\mathrm{id}}_{B},\\ f^{-1}\circ f & = \operatorname {\mathrm{id}}_{A}. \end{align*}
