$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.1.4.5. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if the induced map
\[ \rho : \operatorname{Fun}(\Delta ^2,X) \rightarrow \operatorname{Fun}(\Lambda ^{2}_{1},X) \times _{ \operatorname{Fun}(\Lambda ^{2}_{1},S) } \operatorname{Fun}(\Delta ^2,S) \]
is a trivial Kan fibration.
Proof.
The “only if” direction follows from Proposition 4.1.4.4. For the converse, we observe that $\rho $ is a trivial Kan fibration if and only if $q$ is weakly right orthogonal to the inclusion map
\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2 \]
for every nonnegative integer $m$. Since the collection of inner anodyne morphisms is generated (as a weakly saturated class) by such inclusions (Lemma 1.5.6.9), it follows that $q$ is weakly right orthogonal to all inner anodyne morphisms (Proposition 1.5.4.13) and is therefore an inner fibration (Proposition 4.1.3.1).
$\square$