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Proposition 5.5.5.1. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$ be a functor of $\infty $-categories. Then there is a unique functor of $\infty $-categories

\[ T: \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F} \rightarrow \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}} ) \]

which is the identity on objects and which carries each morphism $(f,u)$ of $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ to the pair $(f, [u])$, regarded as a morphism in the ordinary category $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}$. Moreover, the functor $T$ exhibits the classical Grothendieck construction $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F} }$ as the homotopy category of the $\infty $-categorical Grothendieck construction $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$.

Proof of Proposition 5.5.5.1. We first prove the existence of the functor $T$ appearing in the statement of Proposition 5.5.5.1 (the uniqueness is immediate). Since the induced functor of $2$-categories $\mathrm{h} \mathit{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ is strictly unitary, the construction $(f,u) \mapsto (f,[u])$ carries degenerate edges of the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ to identity morphisms in the category $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}$. It will therefore suffice to show that, for every $2$-simplex $\sigma $ of the simplicial set $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ whose boundary is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & (D,Y) \ar [dr]^{ (g,v) } & \\ (C,X) \ar [ur]^{ (f,u) } \ar [rr]^{(h,w) } & & (E,Z), } \]

we have an identity $(h,[w]) = (g,[v]) \circ (f,[u] )$ in the category $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Note that the functor $\mathscr {F}$ determines a natural isomorphism $\mu : \mathscr {F}(g) \circ \mathscr {F}(f) \xrightarrow {\sim } \mathscr {F}(h)$ in the $\infty $-category $\operatorname{Fun}( \mathscr {F}(C), \mathscr {F}(E) )$. Unwinding the definitions, we see that the composition $(g,[v]) \circ (f,[u] )$ is equal to $( h, [v] \circ [ \mathscr {F}(f)(u) ]\circ [\mu (X)]^{-1} )$. We are therefore reduced to proving the commutativity of the diagram

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

in the homotopy category $\mathrm{h} \mathit{ \mathscr {F}(Z) }$. This commutativity is witnessed by the existence of a diagram

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

in the $\infty $-category $\mathscr {F}(Z)$ itself, which is supplied by the datum of the $2$-simplex $\sigma $ (see Example 5.5.4.15). This completes the construction of the functor $T$.

It follows immediately from the definitions that the functor $T$ is bijective at the level of objects and that, for every pair of objects $(C,X)$ and $(D,Y)$, the induced map

\[ \theta : \pi _0( \operatorname{Hom}_{ \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F} }( (C,X), (D,Y) ) \rightarrow \operatorname{Hom}_{ \int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}} }( (C,X), (D,Y) ) \]

is surjective. To complete the proof, we must show that $\theta $ is also injective. Fix a pair of morphisms $(f,u): (C,X) \rightarrow (D,Y)$ and $(f',u'): (C,X) \rightarrow (D,Y)$ in the $\infty $-category $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ having the same image under $T$, so that $f = f'$ as elements of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ and the morphisms $u,u': \mathscr {F}(f)(X) \rightarrow Y$ are homotopic in the $\infty $-category $\mathscr {F}(D)$. By virtue of Corollary 1.3.3.7, there exists a morphism of simplicial sets $\theta : \operatorname{\raise {0.1ex}{\square }}^2 \rightarrow \mathscr {F}(D)$ whose restriction to the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}(f)(X) \ar [r]^-{\operatorname{id}} \ar [d]^-{ u } & \mathscr {F}(f)(X) \ar [d]^-{u'} \\ Y \ar [r]^-{ \operatorname{id}} & Y. } \]

By virtue of Example 5.5.4.15, $\theta $ determines a $2$-simplex of the $\infty $-category $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ whose boundary is indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & (D,Y) \ar [dr]^{( \operatorname{id}_ D, \operatorname{id}_ Y) } & \\ (C,X) \ar [ur]^{ (f,u) } \ar [rr]^{ (f', u' )} & & (D,Y), } \]

which we can regard as a homotopy from $(f,u)$ to $(f',u')$. $\square$