Kerodon

Proof. Let $\lambda = ( \lambda _{-}, \lambda _{+} ): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ denote the twisted arrow coupling of Example 8.2.0.2, and let $V: \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ denote the composition of $\lambda _{+}$ with projection onto the second factor. Note that $V$ factors as a composition

where the first map is a left fibration (since it is a pullback of $\lambda $, which is a left fibration by virtue of Proposition 8.1.1.15), and the last map is a cocartesian fibration (since it is a pullback of the projection map $\operatorname{\mathcal{E}}^{\dagger } \rightarrow \Delta ^0$). It follows that $V$ is a cocartesian fibration, and that a morphism of $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ is $V$-cocartesian if and only if its image in $\operatorname{\mathcal{E}}^{\dagger }$ is an isomorphism. In particular, our hypotheses on $T$ guarantees that it carries $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Pulling back along the map $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, we obtain a commutative diagram

where the vertical maps are cocartesian fibrations and $T'$ carries $V'$-cocartesian edges to $U'$-cocartesian edges.

Fix a vertex $C \in \operatorname{\mathcal{C}}'$, and let $\operatorname{\mathcal{E}}^{\dagger }(C)$ denote the fiber $V'^{-1} \{ C\} = \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $, so we have a projection map $U^{\dagger }_{C}: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ C\} $. Here $U^{\dagger }_{C}$ is a pullback of $U^{\dagger }$, and therefore a cartesian fibration. Moreover, a morphism of $\operatorname{\mathcal{E}}^{\dagger }(C)$ is $U^{\dagger }_{C}$-cartesian if and only if its image in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian (Remark 5.1.4.6). Let $W_ C$ denote the collection of morphisms of $\operatorname{\mathcal{E}}^{\dagger }(C)$ which satisfy this condition, so that $W = \bigcup _{C \in \operatorname{\mathcal{C}}'} W_ C$. Note that $T'$ restricts to a functor $T_ C: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \{ C\} $. By virtue of Proposition 6.3.4.2, it will suffice to verify the following (for each object $C \in \operatorname{\mathcal{C}}'$):

$(\ast _ C)$

The functor $T_{C}$ exhibits the $\infty $-category $\operatorname{\mathcal{E}}_{C}$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W_{C}$.

In what follows, let us abuse notation by identifying $C$ with its image in the $\infty $-category $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{K}}$ denote the full subcategory of $\operatorname{Tw}( \operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ whose objects are isomorphisms $D \rightarrow C$. By virtue of Proposition 8.1.2.1, $\operatorname{\mathcal{K}}$ can also be described as the full subcategory of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ spanned by its initial objects. It follows that $\operatorname{\mathcal{K}}$ is a coreflective subcategory of $\operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} $ (Example 6.2.2.10). Let $\operatorname{\mathcal{E}}^{\dagger }_0(C) \subseteq \operatorname{\mathcal{E}}^{\dagger }(C)$ denote the inverse image of $\operatorname{\mathcal{K}}$ under $U^{\dagger }_{C}$, so that $\operatorname{\mathcal{E}}^{\dagger }_0(C)$ is a coreflective subcategory of $\operatorname{\mathcal{E}}^{\dagger }(C)$ (Proposition 6.2.4.2). Using Lemma 6.2.2.22, we can choose a functor $L: \operatorname{\mathcal{E}}^{\dagger }(C) \rightarrow \operatorname{\mathcal{E}}^{\dagger }_0(C)$ and a natural transformation $\epsilon : L \rightarrow \operatorname{id}_{ \operatorname{\mathcal{E}}^{\dagger }(C)}$ which exhibits $L$ as a $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$-coreflection functor. Our assumption on $T$ guarantees that the functor $T^{C}$ carries each element of $W_{C}$ to an isomorphism in $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$, so that $\epsilon $ induces an isomorphism of functors $(T^{C}|_{ \operatorname{\mathcal{E}}^{\dagger }_{0}(C)} \circ L) \rightarrow T^ C$. Since the Kan complex $\operatorname{\mathcal{K}}$ is contractible (Corollary 4.7.3.14), the inclusion map $\{ \operatorname{id}_{C} \} \hookrightarrow \operatorname{\mathcal{K}}$ is a homotopy equivalence of Kan complexes, and therefore induces an equivalence of $\infty $-categories $\operatorname{\mathcal{E}}^{\dagger }(C) \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C\} \simeq \operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ (Corollary 4.5.3.33). Since the composition

is an equivalence of $\infty $-categories, we conclude that the functor $T^{C}|_{ \operatorname{\mathcal{E}}^{\dagger }_{0}(C) }$ is also an equivalence of $\infty $-categories. To complete the proof of $(2_ C)$, it will suffice to show that the functor $L$ exhibits $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W_ C$ (see Remark 6.3.1.20). Let $W^+_{C}$ denote the collection of morphisms $v$ of $\operatorname{\mathcal{E}}^{\dagger }(C)$ such that $L(v)$ is an isomorphism in $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$. By virtue of the preceding arguments, this is equivalent to the requirement that $T(v)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{C}$; in particular, assumption $(1)$ guarantees that $W_ C$ is contained in $W^{+}_{C}$. Conversely, if $u: Y \rightarrow Z$ is a morphism of $\operatorname{\mathcal{E}}^{\dagger }(C)$ which belongs to $W^{+}_ C$, then we can choose a commutative diagram

where $u$ and $w$ exhibit $X$ as $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$-coreflections of the objects $Y$ and $Z$, respectively, and therefore belong to $W_{C}$. We are therefore reduced to showing that the functor $L$ exhibits $\operatorname{\mathcal{E}}^{\dagger }_{0}(C)$ as a localization of $\operatorname{\mathcal{E}}^{\dagger }(C)$ with respect to $W^{+}_{C}$, which is a special case of Example 6.3.3.11. $\square$

Proof. We will show that $(2)$ implies $(1)$; the reverse implication is a special case of Proposition 8.6.9.1. Using Corollary 8.6.6.3, we can choose a cocartesian fibration $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ and a commutative diagram

which exhibits $U^{\dagger }$ as a cartesian conjugate of $U'$. Assume that condition $(2)$ is satisfied, so that we have a commutative diagram

where the horizontal maps are equivalences of $\infty $-categories and the vertical maps are isofibrations (Corollary 4.4.5.6). Applying Corollary 4.5.3.31, we deduce that the map

is fully faithful, and that its essential image consists of those functors $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}'$ which carry each morphism of $W$ to an isomorphism in $\operatorname{\mathcal{E}}'$. We may therefore assume without loss of generality that $T' = F \circ T$ for some functor $F \in \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}')$. Proposition 8.6.9.1 implies that $T'$ exhibits $\operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ with respect to $W$. It follows that $F$ is an equivalence of $\infty $-categories (Remark 6.3.1.20), so that $T$ also exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. $\square$

Proof. We will prove assertions $(1)$, $(2)$, and $(3)$; assertion $(4)$ is then a formal consequence. Applying Proposition 8.6.9.1 to $T_0$ we deduce that the functor $Q$ is fully faithful, and that its essential image is spanned by those objects $G \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ which satisfy the following condition:

$(\ast )$

Let $w = (w',w'')$ be a morphism of $\operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $w'$ is $U^{\dagger }$-cartesian and $w''$ has degenerate image in $\operatorname{\mathcal{C}}$, then $G(w)$ is an isomorphism in $\operatorname{\mathcal{E}}$.

Applying Proposition 8.6.2.8 we see that $Q^{\dagger }$ is also fully faithful, and that its essential image consists of those $G$ which satisfy the following:

$(\ast ^{\dagger })$

Let $w = (w',w'')$ be a morphism of $\operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}})$. If $w'$ is a degenerate edge of $\operatorname{\mathcal{D}}^{\dagger }$, then $G(w)$ is a $V$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

This proves assertion $(1)$.

To prove $(2)$, we must show that an object $F \in \operatorname{Fun}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}$ if and only if the induced map

satisfies condition $(\ast ^{\dagger })$. The “only if” direction follows from our assumption that $T_0$ exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. For the converse, assume that $Q(F) = F \circ T_0$ satisfies condition $(\ast ^{\dagger })$ and let $f: X \rightarrow Y$ be a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$, having image $e: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. Choose an object $X' \in \operatorname{\mathcal{D}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \{ C \} $ such that $T_0(X')$ is isomorphic to $X$ as an object of the $\infty $-category $\{ C \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}$. Let $e_{L}: \operatorname{id}_{C} \rightarrow e$ be the morphism in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ introduced in Example 8.1.3.6, and regard $w = (\operatorname{id}_{X'}, e_{L} )$ as an edge of $\operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$. Our assumption on $T_0$ guarantees that $T_0(w)$ is a $U$-cocartesian edge of $\operatorname{\mathcal{D}}$ lying over $e$, and is therefore isomorphic to $f$ (as an object of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$). It will therefore suffice to show that $(F \circ T_0)(w)$ is a $V$-cocartesian edge of $\operatorname{\mathcal{E}}$, which follows from condition $(\ast ^{\dagger })$.

To prove $(3)$, we must show that an object $F^{\dagger } \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\operatorname{op}} }( \operatorname{\mathcal{D}}^{\dagger }, \operatorname{\mathcal{E}}^{\dagger } )$ carries $U^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{D}}^{\dagger }$ to $V^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger }$ if and only if the induced map

satisfies condition $(\ast ^{\dagger })$. The “only if” direction follows immediately from our assumption that $T$ exhibits $V^{\dagger }$ as a cartesian conjugate of $V$, and the reverse implication follows from the criterion of Remark 8.6.1.14. $\square$

Proof of Theorem 8.6.9.9. Using Corollary 8.6.2.4, we can choose a cartesian fibration $U^{\dagger }: \operatorname{\mathcal{E}}^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and a map $T: \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{E}}$ which exhibits $U^{\dagger }$ as a cartesian conjugate of $U$. Then $\mathscr {F}$ is also a contravariant transport representation for $U^{\dagger }$. Set $\operatorname{\mathcal{M}}= \operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$, so that projection onto the middle factor determines a cartesian fibration $V: \operatorname{\mathcal{M}}\rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ whose contravariant transport representation identifies with the functor

Let $\widetilde{W}$ be the collection of all $V$-cartesian morphisms of $\operatorname{\mathcal{M}}$: that is, morphisms of $\operatorname{\mathcal{M}}$ whose image in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian and whose image in $\operatorname{\mathcal{E}}'$ is $U'$-cartesian. Our assumption on $T$ guarantees that the induced map $T': \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ carries $\widetilde{W}$ into $W$. Moreover, we have the following partial converse:

$(\ast )$

Let $(w,w'): (X,X') \rightarrow (Y,Y')$ be a morphism in $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ which belongs to $W$. Then $(w,w')$ is isomorphic to $T'( \widetilde{w} )$, for some morphism $\widetilde{w} \in \widetilde{W}$.

To prove $(\ast )$, let $f: C \rightarrow D$ be the morphism of $\operatorname{\mathcal{C}}$ given by the common image of $w$ and $w'$. Since $T$ induces an equivalence of $\infty $-categories $T_{C}: \operatorname{\mathcal{E}}^{\dagger }_{C} \rightarrow \operatorname{\mathcal{E}}_{C}$, there exists an object $X^{\dagger } \in \operatorname{\mathcal{E}}^{\dagger }_{C}$ such that $T_ C( X^{\dagger } )$ is isomorphic to $X$. Let $u$ be the morphism of $\operatorname{\mathcal{E}}^{\dagger } \times _{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}})$ given by $( \operatorname{id}_{ X^{\dagger } }, f_{L})$, where $f_{L}: \operatorname{id}_{C} \rightarrow f$ is the morphism in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ described in Example 8.1.3.6. Our assumption on $T$ guarantees that $T(u)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$ whose source is equal to $f$, and is therefore isomorphic to $w$ as an object of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \Delta ^1, \operatorname{\mathcal{E}})$. It follows that $\widetilde{w} = (u, w' )$ is a morphism of $\operatorname{\mathcal{M}}$ which satisfies the requirement of $(\ast )$.

Let $\widetilde{W}_{0}$ be the subset of $\widetilde{W}$ consisting of those morphisms $\widetilde{w}$ of $\operatorname{\mathcal{M}}$ such that the image of $\widetilde{w}$ in $\operatorname{\mathcal{E}}^{\dagger }$ is $U^{\dagger }$-cartesian and the image of $\widetilde{w}$ in $\operatorname{\mathcal{E}}'$ is degenerate. It follows from Proposition 8.6.9.1 that the functor $T'$ exhibits $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{M}}$ with respect to $\widetilde{W}$. Combining this observation with $(\ast )$ and Proposition 6.3.1.22, we conclude that $( \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}' )[ W^{-1} ]$ is a localization of $\operatorname{\mathcal{M}}$ with respect to $\widetilde{W}$, which we can identify with the coend $\operatorname{Coend}( \mathscr {F}, \mathscr {F}' )$ (see Proposition 8.6.8.15). $\square$

Proof. We define a category $\operatorname{{\bf \Delta }}_{\ast }$ as follows:

• The objects of $\operatorname{{\bf \Delta }}_{\ast }$ are pairs $([m], i)$, where $[m] = \{ 0 < 1 < \cdots < m \} $ is an object of $\operatorname{{\bf \Delta }}$ and $i$ is an integer satisfying $0 \leq i \leq m$.

• A morphism from $([m], i)$ to $([n], j)$ in $\operatorname{{\bf \Delta }}_{\ast }$ is a nondecreasing function $\alpha : [m] \rightarrow [n]$ satisfying $\alpha (i) \leq j$.

Alternatively, we describe $\operatorname{{\bf \Delta }}_{\ast }$ as the category of elements of the forgetful functor $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Cat}$ (Definition 5.6.1.1). Then the forgetful functor $\operatorname{{\bf \Delta }}_{\ast } \rightarrow \operatorname{{\bf \Delta }}$ induces a cocartesian fibration $V: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }})$ whose covariant transport representation is the standard simplex $\Delta ^{\bullet }$ (Proposition 5.6.3.4). Moreover, a morphism $\alpha : ([m], i) \rightarrow ([n], j)$ is $V$-cocartesian if and only if $\alpha (i) = j$. Let $\operatorname{{\bf \Delta }}_{\ast }^{\circ }$ denote the full subcategory of $\operatorname{{\bf \Delta }}_{\ast }$ spanned by objects $([m], i)$ where $i = m$. Note that the forgetful functor $\operatorname{{\bf \Delta }}_{\ast } \rightarrow \operatorname{{\bf \Delta }}$ restricts to an isomorphism from $\operatorname{{\bf \Delta }}_{\ast }^{\circ }$ to $\operatorname{{\bf \Delta }}$. Note that $\operatorname{{\bf \Delta }}_{\ast }^{\circ }$ is a coreflective subcategory of $\operatorname{{\bf \Delta }}_{\ast }$: the inclusion $\operatorname{{\bf \Delta }}_{\ast }^{\circ } \hookrightarrow \operatorname{{\bf \Delta }}_{\ast }$ admits a right adjoint, given on objects by the construction $([m], i) \mapsto ([i], i)$.

Let $\overline{\operatorname{\mathcal{E}}}$ denote the fiber product $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } ) \times _{ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}) } \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } )$ and let $\overline{W}$ denote the collection of morphisms $w$ of $\overline{\operatorname{\mathcal{E}}}$ such that the image of $w$ in $\operatorname{\mathcal{E}}$ is $U$-cartesian and the image of $w$ in $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } )$ is $V$-cocartesian. It follows from Theorem 8.6.9.9 that the coend $\operatorname{Coend}( \Delta ^{\bullet }, \mathscr {F} )$ can be identified with the localization $\overline{\operatorname{\mathcal{E}}}[ \overline{W}^{-1} ]$.

Let us abuse notation by identifying $\operatorname{\mathcal{E}}$ with the full subcategory of $\overline{\operatorname{\mathcal{E}}}$ given by the inverse image of $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast }^{\circ } ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\ast } )$. Since $U$ is a cartesian fibration, it follows that $\operatorname{\mathcal{E}}$ is a coreflective subcategory of $\overline{\operatorname{\mathcal{E}}}$ (Proposition 6.2.4.2): that is, the inclusion functor $\iota $ admits a right adjoint $L: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{E}}$. Let $\overline{W}_0$ denote the collection of morphisms $\overline{w}$ in $\overline{\operatorname{\mathcal{E}}}$ such that $L( \overline{w} )$ is an isomorphism in $\operatorname{\mathcal{E}}$. Writing $\overline{w} = (w, \alpha )$, where $w$ is a morphism of $\operatorname{\mathcal{E}}$ and $\alpha : ([m], i) \rightarrow ([n], j)$ is a morphism in $\operatorname{{\bf \Delta }}_{\ast }$, we observe that $\overline{w}$ belongs to $\overline{W}_0$ if and only if $w$ is $U$-cartesian and $\alpha $ restricts to a bijection of $[i] \subseteq [m]$ with $[j] \subseteq [n]$. In particular, $\overline{W}_0$ is contained in $\overline{W}$.

Applying Proposition 6.3.3.7, we deduce that the functor $L$ exhibits $\operatorname{\mathcal{E}}$ as a localization of $\overline{\operatorname{\mathcal{E}}}$ with respect to $\overline{W}_0$. Note that for each morphism $\overline{w}$ of $\overline{W}$, the image $L( \overline{w} )$ belongs to $W$. Conversely, every morphism $w$ of $\operatorname{\mathcal{E}}$ which belongs to $W$ can be regarded as an element $\overline{W}$ (by means of the inclusion $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$), and is therefore isomorphic to an element of $L( \overline{W} )$. Applying Proposition 6.3.1.22, we conclude that $L$ induces an equivalence of localizations $\overline{\operatorname{\mathcal{E}}}[ \overline{W}^{-1} ] \rightarrow \operatorname{\mathcal{E}}[ W^{-1} ]$, so that the coend $\operatorname{Coend}( \Delta ^{\bullet }, \mathscr {F} )$ can also be identified with $\operatorname{\mathcal{E}}[W^{-1}]$. $\square$