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Corollary Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. Then, for every simplicial set $K$, the induced map $g: A \star K \hookrightarrow B \star K$ is also inner anodyne.

Proof. The morphism $g$ factors as a composition

\[ A \star K \xrightarrow {g'} B \coprod _{ A } (A \star K) \xrightarrow {g''} B \star K. \]

The morphism $g'$ is inner anodyne since it is a pushout of $f$, and the morphism $g''$ is inner anodyne by virtue of Proposition It follows that $g = g'' \circ g'$ is also inner anodyne. $\square$