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Corollary 4.6.8.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ and $g: X \rightarrow Z$ be morphisms of $\operatorname{\mathcal{C}}$, which we identify with objects of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$. Then the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{X/} }( f, g)$ can be identified with the homotopy fiber of the composition map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(Y, Z ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ over the vertex $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.

Proof. Using Proposition 4.6.8.16, we can replace the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ with the restriction map $\theta : \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $. The morphism $\theta $ is a left fibration (Corollary 4.3.6.11). Since the left-pinched morphism space $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Z)$ is a Kan complex (Proposition 4.6.5.4), it follows that $\theta $ is a Kan fibration (Corollary 4.4.3.8). In particular, the homotopy fiber of the composition map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(Y, Z ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ over the vertex $g$ can be identified with the fiber

\[ \theta ^{-1} \{ g\} \simeq \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}_{X/} } \{ g\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}^{\mathrm{L}}( f, g ), \]

which is homotopy equivalent to $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}(f,g)$ by virtue of Proposition 4.6.5.9. $\square$