Kerodon

Example 2.2.8.18. Let $\operatorname{\mathcal{C}}$ be an ordinary category, regarded as a $2$-category having only identity $2$-morphisms (Remark 2.2.1.6). Then a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ is an isomorphism in the sense of Definition 2.2.8.17 if and only if it is an isomorphism in the usual sense: that is, if and only if there exists a morphism $g: Y \rightarrow X$ satisfying $g \circ f = \operatorname{id}_{X}$ and $f \circ g = \operatorname{id}_ Y$.