Kerodon

Remark 7.5.2.9. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories and let $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ be the restriction of $\mathscr {F}$ to a subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$. Suppose that the inclusion $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 ) \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is left anodyne (this condition is satisfied, for example, if the inclusion map $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ has a right adjoint: see Corollary 7.2.3.7). Then the restriction map $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}_0)$ is a trivial Kan fibration of $\infty $-categories (see Proposition 5.3.1.21).