$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Lemma Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$. Suppose we are given morphisms of simplicial sets $A \xrightarrow {f} B \xrightarrow {g} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $f$ is right anodyne. Then the induced map $A \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow B \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is right anodyne.

Proof. Without loss of generality, we may assume that $f$ is the inclusion map $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for some $0 < i \leq n$. Using Remark, we can reduce to the case where $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $ and $g$ is the identity map. In this case, Remark supplies a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \times \mathscr {F}(0) \ar [r] \ar [d] & \Lambda ^{n}_{i} \times _{ \Delta ^ n} \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d] \\ \Delta ^ n \times \mathscr {F}(0) \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). } \]

It will therefore suffice to show that the left vertical map is right anodyne, which follows from Proposition $\square$