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Lemma Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and suppose we are given morphisms of simplicial sets $A \xrightarrow {f} B \xrightarrow {g} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $f$ is inner anodyne. Then the induced map

\[ \theta _{g}: A \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow B \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \]

is inner anodyne.

Proof. Let $S$ be the collection of all morphisms of simplicial sets $f: A \rightarrow B$ having the property that, for every morphism $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the map $\theta _{g}$ is inner anodyne. It follows immediately from the definitions that $S$ is weakly saturated (in the sense of Definition Consequently, to show that every inner anodyne morphism belongs to $S$, it will suffice to prove that $S$ contained every inner horn inclusion $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. Using Remark, we can reduce to the case where $\operatorname{\mathcal{C}}= [n]$ and $g: \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the identity map. In this case, Remark shows that $\theta _{g}$ is a pushout of the inclusion map $\Lambda ^{n}_{i} \times \mathscr {F}(0) \hookrightarrow \Delta ^ n \times \mathscr {F}(0)$, which is inner anodyne by virtue of Lemma $\square$