Proposition 5.6.3.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set equipped with a morphism $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname{Set})$, which we can identify with a functor $\mathrm{h} \mathit{\mathscr {F}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$. Then the isomorphism of simplicial sets
\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \simeq \operatorname{\mathcal{C}}\times _{ \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) } \operatorname{N}_{\bullet }( \int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathrm{h} \mathit{\mathscr {F}} ) \]
of Example 5.6.2.8 induces an isomorphism of categories
\[ \mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} } \rightarrow \int _{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\mathscr {F}}. \]
Proof.
Using Proposition 4.1.3.2, we can factor the unit map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ as a composition
\[ \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{C}}' \xrightarrow {G} \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ) \]
where $F$ is inner anodyne and $G$ is an inner fibration of simplicial sets (so that $\operatorname{\mathcal{C}}'$ is an $\infty $-category). It follows that $\mathscr {F}$ extends uniquely to a morphism $\mathscr {F}': \operatorname{\mathcal{C}}' \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Using Remark 5.6.2.17, we obtain a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d]^{U} \ar [r]^-{\widetilde{F}} & \int _{\operatorname{\mathcal{C}}'} \mathscr {F}' \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{F} & \operatorname{\mathcal{C}}', } \]
where the vertical maps are cocartesian fibrations (Proposition 5.6.2.2). Since $F$ is inner anodyne, the map $\widetilde{F}$ is a categorical equivalence of simplicial sets (Proposition 5.3.6.1). Moreover, since $F$ is bijective at the level of vertices, $\widetilde{F}$ is also bijective at the level of vertices. It follows that $F$ and $\widetilde{F}$ induce isomorphisms of homotopy categories
\[ \mathrm{h} \mathit{F}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}' \quad \quad \mathrm{h} \mathit{\widetilde{F}'}: \mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} } \rightarrow \mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}'} \mathscr {F}'}. \]
Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$, we are reduced to proving Proposition 5.6.3.2 in the special case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.
Let $\operatorname{\mathcal{D}}$ be the category of elements $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathrm{h} \mathit{\mathscr {F}}$, so that Example 5.6.2.8 supplies a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [d]^{U} \ar [r]^-{G} & \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\overline{G}} & \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}). } \]
We wish to show that $G$ exhibits $\operatorname{\mathcal{D}}$ as a homotopy category of the $\infty $-category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Note that, since $\overline{G}$ is bijective at the level of vertices, the functor $G$ has the same property. It will therefore suffice to show that, for every pair of objects $X,Y \in \int _{\operatorname{\mathcal{C}}} \mathscr {F}$, the induced map
\[ \theta _{X,Y}: \operatorname{Hom}_{\int _{\operatorname{\mathcal{C}}} \mathscr {F} }(X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( G(X), G(Y) ) \]
exhibits $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(G(X), G(Y) )$ as the set of connected components of the Kan complex $\operatorname{Hom}_{\int _{\operatorname{\mathcal{C}}} \mathscr {F}}(X,Y)$. Equivalently, we wish to show that each fiber of the map $\theta _{X,Y}$ is a connected (and therefore nonempty) Kan complex. This is clear, since $\theta _{X,Y}$ is a pullback of the map
\[ \overline{\theta }_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) ) ) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }( U(X), U(Y) ), \]
whose fibers are the connected components of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) )$.
$\square$