# Kerodon

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### 5.7.3 Comparison with the Category of Elements

Let $\mathbf{Cat}$ denote the $2$-category of small categories (Example 2.2.0.4) and let $\operatorname{ \pmb {\mathcal{QC}} }$ denote the $(\infty ,2)$-category of small $\infty$-categories (Construction 5.6.5.1). Suppose we are given a category $\operatorname{\mathcal{C}}$ equipped with a functor $\mathscr {F}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{ \pmb {\mathcal{QC}} }$. Composing with the functor

$\operatorname{ \pmb {\mathcal{QC}} }\rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \mathbf{Cat} ) \quad \quad \operatorname{\mathcal{C}}\mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$

of Remark 5.6.5.8 and invoking Corollary 2.3.4.5, we obtain a (strictly unitary) functor of $2$-categories $\mathrm{h} \mathit{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$, which carries each object $C \in \operatorname{\mathcal{C}}$ to the homotopy category of the $\infty$-category $\mathscr {F}(C)$. Our goal in this section is to compare the $\infty$-category $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ of Definition 5.7.2.1 with the ordinary category $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}$ of Definition 5.7.1.1. We begin with two simple observations:

• Objects of the $\infty$-category $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ can be identified with pairs $(C,X)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $X$ is an object of the $\infty$-category $\mathscr {F}(C)$ (Example 5.7.2.12). Since the $\infty$-category $\mathscr {F}(C)$ and its homotopy category $\mathrm{h} \mathit{\mathscr {F(C)}}$ have the same objects, we can also identify such pairs with objects of the ordinary category $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}$.

• Let $(C,X)$ and $(D,Y)$ be objects of the $\infty$-category $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$. By definition, morphisms from $(C,X)$ to $(D,Y)$ in the $\infty$-category $\int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$ can be identified with pairs $(f, u)$, where $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: \mathscr {F}(f)(X) \rightarrow Y$ is a morphism in the $\infty$-category $\mathscr {F}(D)$ (Example 5.7.2.13). Every such pair determines a morphism $(f, [u])$ in the ordinary category $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}$, where $[u]$ denotes the homotopy class of $u$ (regarded as a morphism in the homotopy category $\mathrm{h} \mathit{\mathscr {F}(D)}$).

Proposition 5.7.3.1. 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 category of elements $\int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F} }$ as the homotopy category of the $\infty$-category of elements $\int _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F}$.

Stated more informally, Proposition 5.7.3.1 asserts that there is a canonical isomorphism of categories

$\mathrm{h} \mathit{ \int _{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \mathscr {F} } \xrightarrow {\sim } \int _{\operatorname{\mathcal{C}}} \mathrm{h} \mathit{\mathscr {F}}.$

In other words, passage to the homotopy category intertwines the classical category of elements construction (Definition 5.7.1.1) with the $\infty$-category of elements construction introduced in §5.7.2.

Proof of Proposition 5.7.3.1. We first prove the existence of the functor $T$ appearing in the statement of Proposition 5.7.3.1 (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 5.7.2.15). 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 5.7.2.15, $\theta$ 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 $(f',u')$. $\square$

In the statement of Proposition 5.7.3.1, it is essential that the source of the functor $\mathscr {F}$ is (the nerve of) an ordinary category. For a more general functor of $\infty$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, one cannot expect to obtain the homotopy category of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ from the construction of Definition 5.7.1.1, because the forgetful functor $\mathrm{h} \mathit{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} } \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ need not be a cocartesian fibration. However, this difficulty does not arise in the case where $\mathscr {F}$ is a set-valued functor:

Proposition 5.7.3.2. 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 5.7.2.8 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 5.7.2.17, 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 5.7.2.2). 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 5.7.3.2 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 5.7.2.8 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 $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Y) )$. $\square$

Corollary 5.7.3.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then $U$ is a left covering map (in the sense of Definition 4.2.3.8) if and only if the following pair of conditions is satisfied:

$(1)$

The induced map $\mathrm{h} \mathit{U}: \mathrm{h} \mathit{\operatorname{\mathcal{E}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a left covering functor (in the sense of Definition 4.2.3.1).

$(2)$

The diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{E}}} ) \ar [d]^{\operatorname{N}_{\bullet }( \mathrm{h} \mathit{U} )} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) }$

is a pullback square.

Proof. The sufficiency of conditions $(1)$ and $(2)$ follows from Proposition 4.2.3.16 and Remark 4.2.3.15. To prove the converse, assume that $U$ is a left covering map. By virtue of Corollary 5.2.7.4, we may assume that $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ for some morphism of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{Set})$. Let us abuse notation by identifying $\mathscr {F}$ with a functor from the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ to the category of sets. Using Proposition 5.7.3.2, we can identify $\mathrm{h} \mathit{\operatorname{\mathcal{E}}}$ with the category of elements $\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathscr {F}$ of Construction 5.2.6.1. Condition $(1)$ now follows from Remark 5.2.6.9, and condition $(2)$ by combining Example 5.7.2.8 with Remark 5.7.2.17. $\square$

We now consider a variant of the situation described in Proposition 5.7.3.1. Let $\operatorname{\mathcal{C}}$ be an ordinary category and suppose we are given a strictly unitary functor of $2$-categories $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$. Passing to the Duskin nerve (and using Remark 5.6.5.7 to identify $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathbf{\operatorname{Cat}} )$ with a full subcategory of $\operatorname{ \pmb {\mathcal{QC}} }$), we obtain a functor of $\infty$-categories $\operatorname{N}_{\bullet }^{\operatorname{D}}( \mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$. Identifying $\mathrm{h} \mathit{\operatorname{N}_{\bullet }^{\operatorname{D}}(\mathscr {F})}$ with the original functor $\mathscr {F}$, Proposition 5.7.3.1 yields a comparison functor

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

Proposition 5.7.3.4. 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.

Stated more informally, Proposition 5.7.3.4 asserts that we can regard the classical category of elements construction (Definition 5.7.1.1) as a special case of Definition 5.7.2.4.

Proof of Proposition 5.7.3.4. By virtue of Proposition 5.7.3.1, 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 1.2.4.1. 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 1.2.4.1 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

5.59
\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 5.7.2.15), 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 (5.59) is automatic (since the projection map $\pi$ is a cocartesian fibration; see Proposition 5.7.2.2). It will therefore suffice to show that $\sigma$ is unique. Using Lemma 4.3.6.14 and Remark 5.6.5.7, we can rewrite (5.59) 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 $\Delta ^{n+1}$. $\square$

Corollary 5.7.3.5. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a morphism of simplicial sets which factors through the full subcategory $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}( \mathbf{Cat} ) ) \subset \operatorname{\mathcal{QC}}$ of Remark 5.6.4.9. Then the projection map $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is an inner covering map of simplicial sets.

Proof. By virtue of Corollary 4.1.5.11, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex. In this case, we wish to show that the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ is isomorphic to the nerve of an ordinary category (see Proposition 4.1.5.10), which is a special case of Proposition 5.7.3.4. $\square$