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Proposition 4.3.6.4 (Joyal [MR1935979]).] Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets. If $f$ is right anodyne or $g$ is left anodyne, then the pushout-join

\[ (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B' \]

is an inner anodyne morphism of simplicial sets.

Proof of Proposition 4.3.6.4. For every pair of morphisms of simplicial sets $f: A \rightarrow A'$ and $g: B \rightarrow B'$, let

\[ \theta _{f,g}: (A \star B') \coprod _{ (A \star B) } (A' \star B) \rightarrow A' \star B' \]

denote their pushout join. We will show that, if $f$ is right anodyne and $g$ is a monomorphism, then $\theta _{f,g}$ is inner anodyne (the analogous assertion for the case where $g$ is left anodyne follows by a similar argument). Let us first regard $f$ as fixed, and let $T$ be the collection of all morphisms $g$ of simplicial sets for which $\theta _{f,g}$ is inner anodyne. Then $T$ is weakly saturated (in the sense of Definition 1.4.4.15). We wish to prove that $T$ contains every monomorphism of simplicial sets. By virtue of Proposition 1.4.5.13, we are reduced to proving that the morphism $\theta _{f,g}$ is inner anodyne in the special case where $g$ is the boundary inclusion $\operatorname{\partial \Delta }^{q} \hookrightarrow \Delta ^{q}$ for some $q \geq 0$.

Let us now regard $g: \operatorname{\partial \Delta }^ q \hookrightarrow \Delta ^ q$ as fixed, and let $S$ denote the collection of all morphisms of simplicial sets for which $\theta _{f,g}$ is inner anodyne. To complete the proof, we must show that $S$ contains every right anodyne morphism of simplicial sets. As before, we note that $S$ is weakly saturated. It will therefore suffice to show that $S$ contains every horn inclusion $\Lambda ^{p}_{i} \hookrightarrow \Delta ^ p$ for $0 < i \leq p$ (see Variant 4.2.4.2). In other words, we are reduced to checking that the pushout-join

\[ \theta _{f,g}: (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]

is inner anodyne. This is clear, since $\theta _{f,g}$ can be identified with the inner horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$ by virtue of Lemma 4.3.6.12. $\square$