Lemma 1.5.7.5. Let $f: X \hookrightarrow Y$ and $f': X' \hookrightarrow Y'$ be monomorphisms of simplicial sets. If $f$ is inner anodyne, then the induced map

is inner anodyne.

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Lemma 1.5.7.5. Let $f: X \hookrightarrow Y$ and $f': X' \hookrightarrow Y'$ be monomorphisms of simplicial sets. If $f$ is inner anodyne, then the induced map

\[ u_{f,f'}: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \hookrightarrow Y \times Y' \]

is inner anodyne.

**Proof.**
Let us regard the morphism $f': X' \hookrightarrow Y'$ as fixed. Let $T$ be the collection of all morphisms $f: X \rightarrow Y$ for which the map $u_{f,f'}$ is inner anodyne. Then $T$ is weakly saturated. To prove Lemma 1.5.7.5, we must show that $T$ contains all inner anodyne morphisms of simplicial sets. By virtue of Lemma 1.5.6.9, it will suffice to show that $T$ contains every morphism of the form

\[ u_{i,j}: (B \times \Lambda ^2_1) \coprod _{A \times \Lambda ^2_1 } (A \times \Delta ^2) \subseteq B \times \Delta ^2, \]

where $i: A \hookrightarrow B$ is a monomorphism of simplicial sets and $j: \Lambda ^2_1 \hookrightarrow \Delta ^2$ is the inclusion. Setting

\[ A' = (B \times X') \coprod _{ (A \times X' )} ( A \times Y') \quad \quad B' = B \times Y', \]

we are reduced to the problem of showing that the map

\[ u_{i',j}: (B' \times \Lambda ^2_1) \coprod _{A' \times \Lambda ^2_1 } (A' \times \Delta ^2) \subseteq B' \times \Delta ^2, \]

is inner anodyne, which follows from Lemma 1.5.6.9. $\square$