Kerodon

Example 7.4.2.8. Let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ be the nerve of a category $\operatorname{\mathcal{C}}_0$. Suppose that $\mathscr {F}, \mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ are obtained as the nerves of strictly commutative diagrams $\mathscr {F}_0, \mathscr {F}'_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Kan}$, and let $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ be obtained from a natural transformation $\gamma _0: \mathscr {F}_0 \rightarrow \mathscr {F}'_0$. Applying the weighted nerve construction to $\gamma _0$, we obtain a functor $\Gamma : \operatorname{N}_{\bullet }^{\mathscr {F}_0}(\operatorname{\mathcal{C}}_0) \rightarrow \operatorname{N}_{\bullet }^{ \mathscr {F}'_0 }(\operatorname{\mathcal{C}}_0)$ which is compatible with $\gamma $.