Remark 2.2.8.22. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be $1$-morphisms of $\operatorname{\mathcal{C}}$. If any two of the $1$-morphisms $f$, $g$, and $g \circ f$ is an isomorphism, then so is the third. In particular, the collection of isomorphisms is closed under composition.
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