Remark 5.3.4.14 (0376): Functoriality

Remark 5.3.4.14 (Functoriality). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the morphism of Construction 5.3.4.11. If $T: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ is any functor between categories, then $\lambda $ induces a morphism

\[ \lambda '_{t}: \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}). \]

Setting $\mathscr {F}' = \mathscr {F} \circ T$, we can use Remarks 5.3.2.3 and 5.3.3.7 to identify $\lambda '_{t}$ with a morphism from the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' )$ to the weighted nerve $\operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}')$. This morphism coincides with the map obtained by applying Construction 5.3.4.11 to the diagram $\mathscr {F}'$.