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Proposition 5.5.5.3. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a strictly unitary functor of $2$-categories. Then the comparison map

\[ T: \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \]

is an isomorphism of simplicial sets.

Proof of Proposition 5.5.5.3. By virtue of Proposition 5.5.5.1, it will suffice to show that the simplicial set $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ is isomorphic to the nerve of a category. We will prove this by verifying the criterion of Proposition 1.2.3.1. Fix $0 < i < n$; we wish to show that every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$ can be extended uniquely to an $n$-simplex $\sigma $ of $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$. Let $\overline{\sigma }_0$ denote the composition of $\sigma _0$ with the projection map $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{D}}(\mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Proposition 1.2.3.1 then guarantees that $\overline{\sigma }_0$ extends uniquely to a morphism of simplicial sets $\overline{\sigma }: \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. It will therefore suffice to show that the lifting problem

5.50
\begin{equation} \begin{gathered}\label{equation:Grothendieck-of-ordinary} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} ) \ar [d]^-{\pi } \\ \Delta ^ n \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{ \sigma } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \end{gathered} \end{equation}

has a unique solution.

We begin by treating the special case $n=2$ (so that $i=1$). In this case, we can identify $\sigma _0$ with a pair of composable morphisms

\[ (C,X) \xrightarrow { (f,u) } (D,Y) \xrightarrow { (g,v) } (E,Z) \]

in the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F} )$. Set $h = g \circ f \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,E)$, so that the composition constraint of $\mathscr {F}$ determines an isomorphism of functors $\mu : \mathscr {F}(g) \circ \mathscr {F}(f) \xrightarrow {\sim } \mathscr {F}(h)$. Unwinding the definitions (using Example 5.5.4.15), we are reduced to proving that there is a unique morphism $w: \mathscr {F}(h)(X) \rightarrow Z$ in the category $\mathscr {F}(E)$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ (\mathscr {F}(g) \circ \mathscr {F}(f))(X) \ar [r]^-{ \mu (X) } \ar [d]^-{\mathscr {F}(u)} & \mathscr {F}(h)(X) \ar [d]^-{ w } \\ \mathscr {F}(g)(Y) \ar [r]^-{v } & Z } \]

commutes. This is clear, since $\mu (X)$ is an isomorphism in the category $\mathscr {F}(E)$.

We now treat the case $n \geq 3$. Note that the existence of a solution to the lifting problem (5.50) is automatic (since the projection map $\pi $ is a cocartesian fibration; see Proposition 5.5.4.2). It will therefore suffice to show that $\sigma $ is unique. Using the definition of the Grothendieck construction, Lemma 4.3.6.14, and Remark 5.4.5.7, we can rewrite (5.50) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i+1} \ar [r]^-{\tau _0} \ar [d] & \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathbf{Cat} ) \ar [d] \\ \Delta ^{n+1} \ar [r] \ar@ {-->}[ur]^{\tau } & \Delta ^{0}. } \]

The uniqueness of its solution is now an immediate consequence of Proposition 2.3.1.9, since the horn $\Lambda ^{n+1}_{i+1}$ contains the $2$-skeleton of $\Delta ^{n+1}$. $\square$