Corollary 126.96.36.199. 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
The morphism $g'$ is inner anodyne since it is a pushout of $f$, and the morphism $g''$ is inner anodyne by virtue of Proposition 188.8.131.52. It follows that $g = g'' \circ g'$ is also inner anodyne. $\square$