Exercise 2.2.6.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strictly unitary isomorphism of $2$-categories. Show that there is a unique twisting cochain $\{ \mu _{g,f} \} $ on the $2$-category $\operatorname{\mathcal{C}}$ such that $F$ factors as a composition $\operatorname{\mathcal{C}}\xrightarrow {G} \operatorname{\mathcal{C}}' \xrightarrow {H} \operatorname{\mathcal{D}}$, where $G$ is the strictly unitary isomorphism of Exercise 2.2.6.9 and $H$ is a strict isomorphism of $2$-categories. In other words, the notion of twisting cochain (in the sense of Notation 2.2.6.7) measures the difference between strictly unitary isomorphisms and strict isomorphisms in the setting of $2$-categories.

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