Exercise 2.2.6.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category equipped with a twisting cochain $\{ \mu _{g,f} \} $. Show that the $2$-category $\operatorname{\mathcal{C}}'$ of Construction 2.2.6.8 is well-defined. Moreover, there is a strictly unitary isomorphism of $2$-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ which carries each object, $1$-morphism, and $2$-morphism of $\operatorname{\mathcal{C}}$ to itself, where the composition constraints are given by $\{ \mu _{g,f} \} $.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$