Kerodon

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Proposition 4.6.1.18. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: q(X) \rightarrow q(Y)$ be an edge of $\operatorname{\mathcal{D}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e}$ is a Kan complex.

Proof. Form a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{q'} & \operatorname{\mathcal{C}}\ar [d]^{q} \\ \Delta ^1 \ar [r]^-{e} & \operatorname{\mathcal{D}}. } \]

Since $q$ is an inner fibration, the morphism $q'$ is also an inner fibration (Remark 4.1.1.5), so that $\operatorname{\mathcal{C}}'$ is an $\infty $-category (Remark 4.1.1.9). Remark 4.6.1.17 then supplies an isomorphism of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{e}$ with a simplicial set of the form $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}( \widetilde{X}, \widetilde{Y} )$, which is a Kan complex by virtue of Proposition 4.6.1.9. $\square$