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8.2.2 Profunctors of $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories. A profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ is a $\operatorname{Set}$-valued functor on the product category $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$. This notion has an evident $\infty $-categorical analogue, where we replace the ordinary category of sets by the $\infty $-category $\operatorname{\mathcal{S}}$ of spaces (see Construction 5.5.1.1).

Definition 8.2.2.1. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. A profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ is a functor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$.

Example 8.2.2.2. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. Then every functor $K: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}$ determines a morphism of simplicial sets

\[ \operatorname{N}_{\bullet }(K): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}}. \]

This construction determines a monomorphism from the collection of profunctors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ (in the sense of classical category theory) to the collection of profunctors from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 8.2.2.1). Beware that this map is (usually) not bijective: its image consists of those profunctors

\[ \mathscr {K}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}} \]

having the property that for every pair of objects $X \in \operatorname{\mathcal{C}}$ and $Y \in \operatorname{\mathcal{D}}$, the Kan complex $\mathscr {K}(X,Y)$ is a constant simplicial set (see Proposition 1.2.2.1 and Remark 5.5.1.7).

Remark 8.2.2.3 (Symmetry). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$. Then, by transposing its arguments, we can also regard $\mathscr {K}$ as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{D}}^{\operatorname{op}}$.

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. Every functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ determines a profunctor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}\quad \quad (X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) ). \]

We say that a profunctor is representable if (up to isomorphism) it is obtained in this way. Equivalently, a profunctor $K: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{Set}$ is representable if, for each object $Y \in \operatorname{\mathcal{D}}$, the functor $K(-,Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is representable by an object of $\operatorname{\mathcal{C}}$ (Exercise 8.2.2.9). This condition generalizes to the setting of $\infty $-categories.

Definition 8.2.2.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$. We say that $\mathscr {H}$ is corepresentable if, for each object $X \in \operatorname{\mathcal{C}}$, the functor $\mathscr {H}(X, =): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable, in the sense of Definition 5.6.6.1. We will say that $\mathscr {K}$ is representable if, for each object $Y \in \operatorname{\mathcal{D}}$, the functor $\mathscr {H}(-,Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable, in the sense of Variant 5.6.6.2.

Warning 8.2.2.5. The terminology of Definition 8.2.2.4 is potentially confusing. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, let $\operatorname{\mathcal{E}}$ denote the product $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}$, and let $\mathscr {K}: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}$ be a morphism of simplicial sets. In general, there is no relationship between the corepresentability of $\mathscr {K}$ as a $\operatorname{\mathcal{S}}$-valued functor $\operatorname{\mathcal{E}}$ (in the sense of Definition 5.6.6.1) and the corepresentability of $\mathscr {K}$ as a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.2.2.4). However, these notions of corepresentability coincide when $\operatorname{\mathcal{C}}$ is a contractible Kan complex (see Example 8.2.2.8).

Remark 8.2.2.6. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$. Then $\mathscr {K}$ is representable if and only if it is corepresentable when regarded as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (see Remark 8.2.2.3).

Remark 8.2.2.7 (Symmetry). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\mathscr {K}$ and $\mathscr {K}'$ be profunctors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ which are isomorphic as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$. Then $\mathscr {K}$ is representable if and only if $\mathscr {K}'$ is representable. Similarly, $\mathscr {K}$ is corepresentable if and only if $\mathscr {K}'$ is corepresentable. See Remark 5.6.6.4.

Example 8.2.2.8. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. If $\operatorname{\mathcal{C}}= \Delta ^0$, then the profunctor $\mathscr {K}$ is corepresentable (in the sense of Definition 8.2.2.4) if and only if it is corepresentable when regarded as a functor $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ (in the sense of Definition 5.6.6.1). Similarly, if $\operatorname{\mathcal{D}}= \Delta ^0$, then the profunctor $\mathscr {K}$ is representable (in the sense of Definition 8.2.2.4) if and only if it is representable when viewed as a functor $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ (in the sense of Variant 5.6.6.2).

Exercise 8.2.2.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be ordinary categories. Show that a profunctor

\[ \mathscr {K}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{S}} \]

is representable (in the sense of Definition 8.2.2.4) if and only if it it is isomorphic to the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) )$, for some functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. See Proposition 8.2.6.1 for a more general result.

Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. Then a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ can be identified with a functor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$. We now study conditions which guarantee that this functor is fully faithful.

Proposition 8.2.2.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a corepresentable profunctor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$. The following conditions are equivalent:

$(1)$

The profunctor $\mathscr {K}$ determines a fully faithful functor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, -). \]
$(2)$

Let $X$ be an object of $\operatorname{\mathcal{C}}$, let $Y$ be an object of $\operatorname{\mathcal{D}}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. If $\eta $ exhibits the functor $\mathscr {K}(X, -): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$, then it also exhibits the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$.

Proof. Choose an object $X$ be an object of $\operatorname{\mathcal{C}}$. Then the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable. We can therefore choose an object $Y \in \operatorname{\mathcal{D}}$ and a vertex $\eta \in \mathscr {K}(X,Y)$ which exhibits the functor $\mathscr {K}(X,-)$ as corepresented by $Y$. We will show that the following conditions are equivalent:

$(1_ X)$

For every object $X' \in \operatorname{\mathcal{C}}$, the profunctor $\mathscr {K}$ induces a homotopy equivalence

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}( X, X' ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) }( \mathscr {K}(X, -), \mathscr {K}(X'-) ). \]
$(2_ X)$

The vertex $\eta $ exhibits the functor $\mathscr {K}(-, Y)$ as represented by $X$.

Proposition 8.2.2.10 will then follow by allowing the triple $(X,Y,\eta )$ to vary.

Condition $(2_ X)$ is the assertion that, for each object $X' \in \cal \operatorname{\mathcal{C}}$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}( X, X' ) & \rightarrow & \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {K}(X, -), \mathscr {K}(X', -) ) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {K}(X,Y), \mathscr {K}(X', Y) ) \\ & \xrightarrow { \circ [\eta ] } & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {K}(X',Y) ) \\ & \simeq & \mathscr {K}(X',Y) \end{eqnarray*}

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. The equivalence of this assertion with $(1_ X)$ follows immediately from Proposition 8.2.1.3. $\square$

Definition 8.2.2.11 (Balanced Profunctors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We say that a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is balanced if it satisfies the following conditions:

  • The profunctor $\mathscr {K}$ is representable and corepresentable (Definition 8.2.2.4).

  • Let $X$ be an object of $\operatorname{\mathcal{C}}$, let $Y$ be an object of $\operatorname{\mathcal{D}}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. Then $\eta $ exhibits the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ as represented by $X$ if and only if it exhibits the functor $\mathscr {K}( X, -): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by $Y$.

In other words, $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is balanced if it satisfies the hypotheses of Proposition 8.2.2.10 both when regarded as a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$ and when regarded as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{D}}^{\operatorname{op}}$.

Corollary 8.2.2.12. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The profunctor $\mathscr {K}$ is balanced (in the sense of Definition 8.2.2.11).

$(2)$

The $\infty $-category $\operatorname{\mathcal{D}}$ is locally small and $\mathscr {K}$ induces a fully faithful functor

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ), \]

whose essential image is spanned by the corepresentable functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$.

$(3)$

The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small and $\mathscr {K}$ induces a fully faithful functor

\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad Y \mapsto \mathscr {K}(-, Y ), \]

whose essential image is spanned by the representable functors $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$.

Proof. We will prove the equivalence of $(1)$ and $(2)$; the equivalence of $(1)$ and $(3)$ follows by a similar argument. Assume first that $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is a balanced profunctor. Invoking Proposition 8.2.2.10, we see that the functor

\[ \Phi : \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \quad \quad X \mapsto \mathscr {K}(X, - ) \]

is fully faithful, and that the essential image of $\Phi $ consists of corepresentable functors from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{S}}$. Fix an object $Y \in \operatorname{\mathcal{D}}$. Since $\mathscr {K}$ is representable, there exists an object $X \in \operatorname{\mathcal{C}}$ and a vertex $\eta \in \mathscr {K}(X,Y)$ which exhibits the functor $\mathscr {K}( -, Y)$ as represented by $X$. Our assumption that $\mathscr {K}$ is balanced guarantees that $\eta $ also exhibits the functor $\mathscr {K}(X, -)$ as corepresented by $Y$. In particular, for every object $Y' \in \operatorname{\mathcal{D}}$, $\eta $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(Y,Y') \xrightarrow {\sim } \mathscr {K}(X,Y')$, so that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y')$ is essentially small. If $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is any functor corepresented by $Y$, then Theorem 5.6.6.14 guarantees that $F$ is isomorphic to $\mathscr {K}(X, -)$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$), and therefore belongs to the essential image of $\Phi $. Allowing the object $Y$ to vary, we deduce that the profunctor $\mathscr {K}$ satisfies condition $(2)$.

We now prove the converse. Assume that the functor $\Phi $ is fully faithful and that the essential image of $\Phi $ is spanned by the corepresentable functors $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$. We wish to show that the profunctor $\mathscr {K}$ is balanced. Since $\Phi $ takes values in the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors, the profunctor $\mathscr {K}$ is corepresentable. We next show that $\mathscr {K}$ is representable. Fix an object $Y \in \operatorname{\mathcal{D}}$; we wish to show that the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable. Since $\operatorname{\mathcal{C}}$ is locally small, there exists a functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ which is corepresentable by $Y$ (Theorem 5.6.6.14). Then $F$ belongs to the essential image of $\Phi $. We may therefore assume without loss of generality that $F = \mathscr {K}( X_0, -)$ for some object $X_0 \in \operatorname{\mathcal{C}}$. Choose a vertex $\eta _0 \in \mathscr {K}(X_0,Y) = F(Y)$ which exhibits the functor $F$ as corepresented by $Y$. Since $\Phi $ is fully faithful, Proposition 8.2.2.10 implies that $\eta _0$ also exhibits the functor $\mathscr {K}(-, Y)$ as represented by $X_0$.

To complete the proof, we must show that the pairing $\mathscr {K}$ satisfies the second condition of Definition 8.2.2.11. Let $Y \in \operatorname{\mathcal{D}}$ be as above, let $X$ be any object of $\operatorname{\mathcal{C}}$, and let $\eta $ be a vertex of the Kan complex $\mathscr {K}(X,Y)$. Assume that $\eta $ exhibits the functor $\mathscr {K}(-, Y)$ as represented by $X$; we wish to show that it also exhibits the functor $\mathscr {K}(X, -)$ as corepresented by $Y$ (the reverse implication follows from Proposition 8.2.2.10). Let $\eta _0 \in \mathscr {K}(X_0, Y)$ be as above. Since $\eta _0$ exhibits $\mathscr {K}(-, Y)$ as represented by $X_0$, there exists an isomorphism $u: X \rightarrow X_0$ in the $\infty $-category $\operatorname{\mathcal{C}}$ such that $\mathscr {K}(u,\operatorname{id}_ Y)( \eta _0 )$ and $\eta $ belong to the same connected component of the Kan complex $\mathscr {K}(X,Y)$ (Remark 5.6.6.6). We may therefore assume without loss of generality that $\eta = \mathscr {K}(u, \operatorname{id}_ Y)(\eta _0)$ (Remark 5.6.6.3). The desired result now follows by applying Remark 5.6.6.4 to the isomorphism of functors $\mathscr {K}(u, -): \mathscr {K}(X_0, -) \rightarrow \mathscr {K}(X, -)$. $\square$

Corollary 8.2.2.13. Let $\operatorname{\mathcal{C}}$ be a locally small $\infty $-category, and let $\operatorname{Fun}^{\mathrm{corep}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors. Then the evaluation map

\[ \operatorname{ev}: \operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}\quad \quad (F,C) \mapsto F(C) \]

is a balanced profunctor.

Remark 8.2.2.14 (Uniqueness). Up to equivalence, every balanced profunctor can be obtained from the construction of Corollary 8.2.2.13. More precisely, let $\operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors. If $\mathscr {K}$ is a balanced profunctor from $\operatorname{\mathcal{D}}$ to $\operatorname{\mathcal{C}}$, then it factors as a composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\xrightarrow {\Phi \times \operatorname{id}} \operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{S}}) \times \operatorname{\mathcal{D}}\xrightarrow {\operatorname{ev}} \operatorname{\mathcal{S}}, \]

where $\Phi $ is an equivalence of $\infty $-categories (Corollary 8.2.2.12).