Proposition 4.3.7.1. Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets, and let

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

be the pushout-join of Construction 4.3.6.3. If $f$ is anodyne, then $\theta _{f,g}$ is left anodyne. If $g$ is anodyne, then $\theta _{f,g}$ is right anodyne.

**Proof.**
We will prove the first assertion; the proof of the second is similar. We proceed as in the proof of Proposition 4.3.6.4. Let us first regard the anodyne morphism $f$ as fixed, and let $T$ be the collection of all morphisms $g$ of simplicial sets for which $\theta _{f,g}$ is left anodyne. Then $T$ weakly saturated (in the sense of Definition 1.5.4.12). We wish to prove that $T$ contains every monomorphism of simplicial sets. By virtue of Proposition 1.5.5.14, we are reduced to proving that the morphism $\theta _{f,g}$ is left 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 left anodyne. To complete the proof, we must show that $S$ contains every 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 $\Lambda ^{p}_{i} \hookrightarrow \Delta ^ p$ when $p > 0$ and $0 \leq i \leq p$. 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 left anodyne. This is clear, since $\theta _{f,g}$ can be identified with the horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$ by virtue of Lemma 4.3.6.15.
$\square$