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Lemma Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets, and let $g: B \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set_{\Delta }})$ be any morphism of simplicial sets. Then the induced map

\[ \theta _{g}: A \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \rightarrow B \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \]

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 $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$, 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 $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. In this case, we can identify a morphism $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$ with a diagram of simplicial sets $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$, which we denote by

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

To complete the proof, we must show that the inclusion map $\theta : \Lambda ^{n}_{i} \times _{\Delta ^ n} M( \overrightarrow {X} ) \hookrightarrow M( \overrightarrow {X} )$ is inner anodyne. Using Example, we see that $\theta $ is a pushout of the inclusion $\Lambda ^{n}_{i} \times X(0) \hookrightarrow \Delta ^ n \times X(0)$, which is inner anodyne by virtue of Lemma $\square$