Kerodon

Exercise 2.2.6.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strictly unital isomorphism of bicategories. Show that there is a unique twisting cochain $\{ \mu _{g,f} \} $ on the bicategory $\operatorname{\mathcal{C}}$ 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 unital isomorphism of Exercise 2.2.6.9 and $H$ is a strict isomorphism of bicategories. In other words, the notion of twisting cochain (in the sense of Notation 2.2.6.7) measures the difference between strictly unital isomorphisms and strict isomorphisms in the setting of bicategories.