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} & = \text{id}_{B},\\ f^{-1}\circ f & = \text{id}_{A}. \end{align*}

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