$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 5.4.2.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an interior fibration between $(\infty ,2)$-categories, and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$. Then:
- $(1)$
The induced map of left-pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{L}}( F(X), F(Y) )$ is a right fibration of simplicial sets.
- $(2)$
The induced map of right-pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{R}}( F(X), F(Y) )$ is a left fibration of simplicial sets.
Proof.
We will prove $(2)$; assertion $(1)$ follows from a similar argument. We wish to show that, for every pair of integers $0 \leq i < n$, every lifting problem
5.36
\begin{equation} \begin{gathered}\label{equation:interior-fibration-morphism-space} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \ar [d] \\ \Delta ^{n} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{R}}( F(X), F(Y) ) } \end{gathered} \end{equation}
admits a solution. Unwinding the definitions, we can rewrite (5.36) as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i} \ar [r]^-{\tau _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d]^-{F} \\ \Delta ^{n+1} \ar [r]^-{\overline{\tau }} \ar@ {-->}[ur]^{\tau } & \operatorname{\mathcal{D}}, } \]
where the restriction $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < \cdots < n \} )}$ is the constant map taking the value $X$. If $i=0$, then this lifting problem admits a solution because the edge $\tau _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is degenerate (and therefore $F$-cocartesian, by virtue of our assumption that $F$ is an interior fibration). If $0 < i < n$, the solution exists by virtue of the fact that $F$ is an interior fibration and $\overline{\tau }|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is a degenerate $2$-simplex of $\operatorname{\mathcal{D}}$ (and therefore thin).
$\square$