Exercise 8.1.8.3. Show that Construction 8.1.8.2 is well-defined. That is, given a functor $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as in Construction 8.1.8.2, show that there is a unique strictly unitary lax functor $U_{F}$ satisfying properties $(1)$, $(2)$, and $(3)$ of Construction 8.1.8.2.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$