Lemma 5.4.2.6. Let $\operatorname{\mathcal{D}}$ be an $(\infty ,2)$-category, let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration of simplicial sets, and let $\sigma $ be a $2$-simplex of $\operatorname{\mathcal{C}}$. If $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$, then $\sigma $ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Suppose we are given a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$, where $n \geq 3$, $0 < i < n$, and $\tau _0$ carries $\operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} )$ to the $2$-simplex $\sigma $. We wish to show that $\tau _0$ can be extended to an $n$-simplex $\tau $ of $\operatorname{\mathcal{C}}$. Let $\overline{\tau }_0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{D}}$ be the composition $q \circ \tau _0$. Since $q(\sigma )$ is a thin $2$-simplex of $\operatorname{\mathcal{D}}$, we can extend $\overline{\tau }_0$ to an $n$-simplex $\overline{\tau }: \Delta ^{n} \rightarrow \operatorname{\mathcal{D}}$. To complete the proof, it suffices to find a solution to the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{ \tau _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{q} \\ \Delta ^{n} \ar@ {-->}[ur]^{\tau } \ar [r]^-{\overline{\tau }} & \operatorname{\mathcal{D}}, } \]
which exists by virtue of our assumption that $q$ is an interior fibration. $\square$