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8.2.1 Representable Couplings

We now axiomatize an essential feature of the couplings that can be obtained from Construction 8.2.0.3.

Definition 8.2.1.1. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $C$ be an object of $\operatorname{\mathcal{C}}$, having image $\lambda (C) = (C_{-}, C_{+} ) \in \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$. We say that $X$ is universal if it is an initial object of the $\infty $-category $\operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \{ C_{+} \} $, and couniversal if it is an initial object of the $\infty $-category $\{ C_{-} \} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{C}}$.

Remark 8.2.1.2 (Uniqueness). Let $\lambda = (\lambda _{+}, \lambda _{-}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories, let $C$ be a universal object of $\operatorname{\mathcal{C}}$, and let $D$ be another object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

  • The object $C$ is isomorphic to $D$ (as an object of the $\infty $-category $\operatorname{\mathcal{C}}$).

  • The object $D$ is universal, and $\lambda _{+}(D)$ is isomorphic to $\lambda _{+}(C)$ (as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{+}$).

Definition 8.2.1.3. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories.

  • We say that $\lambda $ is representable if, for every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$, there exists a universal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{+}(C) = C_{+}$.

  • We say that $\lambda $ is corepresentable if, for every object $C_{-} \in \operatorname{\mathcal{C}}_{-}$, there exists a couniversal object $C \in \operatorname{\mathcal{C}}$ satisfying $\lambda _{-}(C) = C_{-}$.

Remark 8.2.1.4 (Symmetry). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories, and let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\operatorname{\mathcal{C}}_{+}$ with $\operatorname{\mathcal{C}}_{-}$. Then the transposition $\lambda ' = ( \lambda _{+}, \lambda _{-} )$ can be regarded as a coupling of $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ with $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$. In this situation:

  • An object $C \in \operatorname{\mathcal{C}}$ is universal for the coupling $\lambda $ if and only if it is couniversal for the coupling $\lambda '$.

  • The coupling $\lambda $ is representable if and only if the coupling $\lambda '$ is corepresentable.

Example 8.2.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda : \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ be the twisted arrow coupling of Example 8.2.0.2. For every morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, Corollary 8.1.2.17 asserts that the following conditions are equivalent:

$(1)$

The morphism $f$ is an isomorphism in $\operatorname{\mathcal{C}}$.

$(2)$

As an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$, $f$ is couniversal with respect to the coupling $\lambda $.

$(3)$

As an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$, $f$ is universal with respect to the coupling $\lambda $.

In particular, the coupling $\lambda $ is both representable and corepresentable.

Variant 8.2.1.6. Let $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ be a functor of $\infty $-categories, set $\operatorname{\mathcal{C}}= \operatorname{Tw}(\operatorname{\mathcal{C}}_{-}) \times _{ \operatorname{\mathcal{C}}_{-} } \operatorname{\mathcal{C}}_{+}$, and let $\lambda _{G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ denote the coupling of Construction 8.2.0.3. Unwinding the definitions, we see that objects of $\operatorname{\mathcal{C}}$ can be identified with pairs $(e, C_{+} )$, where $C_{+}$ is an object of the $\infty $-category $\operatorname{\mathcal{C}}_{+}$ and $e: C_{-} \rightarrow G(C_{+})$ is a morphism in the $\infty $-category $\operatorname{\mathcal{C}}_{-}$. It follows from Example 8.2.1.5 that an object $(e,C_{+}) \in \operatorname{\mathcal{C}}$ is universal if and only if $e$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{-}$. Note that every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$ can be lifted to a universal object of $\operatorname{\mathcal{C}}$ (for example, we can choose $e$ to be the identity morphism from $G(C_{+})$ to itself), so that the coupling $\lambda _{G}$ is representable. In ยง8.2.3, we will prove the converse: every representable coupling of $\infty $-categories can be obtained (up to equivalence) from Construction 8.2.0.3(Theorem 8.2.0.4).

Our goal in this section is to establish a universal mapping property of (co)representable couplings (Proposition 8.2.1.8). First, we give a reformulation of Definition 8.2.0.1 (compare with Corollary 8.1.1.13).

Proposition 8.2.1.7. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. Then a morphism of simplicial sets $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_+$ is a left fibration if and only if it satisfies the following conditions:

$(1)$

The morphism $\lambda $ is an isofibration; in particular, $\operatorname{\mathcal{C}}$ is an $\infty $-category.

$(2)$

The functor $\lambda _{-}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ is a cocartesian fibration. Moreover, a morphism $u$ of $\operatorname{\mathcal{C}}$ is $\lambda _{-}$-cocartesian if and only if $\lambda _{+}(u)$ is an isomorphism in $\operatorname{\mathcal{C}}_{+}$.

$(3)$

The functor $\lambda _{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{+}$ is a cocartesian fibration. Moreover, a morphism $u$ of $\operatorname{\mathcal{C}}$ is $\lambda _{+}$-cocartesian if and only if $\lambda _{-}(u)$ is an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$.

Proof. Suppose first that $\lambda $ is a left fibration. Then $\lambda $ is a cocartesian fibration, and every morphism of $\operatorname{\mathcal{C}}$ is $\lambda $-cocartesian (Proposition 5.1.4.14). In particular, $\lambda $ is an isofibration. Let $\pi _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ and $\pi _{+}: \operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{+}$ denote the projection maps. Note that $\pi _{-}$ is a cocartesian fibration, and that a morphism $(u_-, u_{+})$ of $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ is $\pi _{-}$-cocartesian if and only if $u_{+} = \pi _{+}( u_{-}, u_{+} )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{+}$ (see Remark 5.1.4.6). Applying Proposition 5.1.4.13, we see that $\lambda _{-} = \pi _{-} \circ \lambda $ is also a cocartesian fibration, and that a morphism $u$ of $\operatorname{\mathcal{C}}$ is $\lambda _{-}$-cocartesian if and only if $\pi _{+}( \lambda (u) ) = \lambda _{+}(u)$ is an isomorphism in $\operatorname{\mathcal{C}}_{+}$. This proves assertion $(2)$, and assertion $(3)$ follows by a similar argument.

We now prove the converse. Suppose that $\lambda $ satisfies conditions $(1)$, $(2)$, and $(3)$; we wish to show that $\lambda $ is a left fibration. We first show that $\lambda $ is a cocartesian fibration. Fix an object $X \in \operatorname{\mathcal{C}}$ having image $\overline{X} = \lambda (X)$, together with a morphism $\overline{w}: \overline{X} \rightarrow \overline{Z}$ in the product $\operatorname{\mathcal{C}}_{-} \times \operatorname{\mathcal{C}}_{+}$. We wish to show that we can write $\overline{u} = \lambda (w)$ for some $\lambda $-cocartesian morphism $w: X \rightarrow Z$ in $\operatorname{\mathcal{C}}$. Invoking assumption $(2)$, we can choose a $\lambda _{-}$-cocartesian morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ satisfying $\lambda _{-}(u) = \pi _{-}(\overline{w} )$. Set $\overline{Y} = \lambda (Y)$ and $\overline{u} = \lambda (u)$. Note that the morphism $\lambda _{+}(u) = \pi _{+}( \overline{u} )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{+}$. We can therefore choose a $2$-simplex $\overline{\sigma }$ of $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$ as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \overline{Y} \ar [dr]^-{ \overline{v} } & \\ \overline{X} \ar [ur]^{ \overline{u} } \ar [rr]^{ \overline{w} } & & \overline{Z}, } \]

for which the image $\pi _{-}( \overline{\sigma } )$ is a right-degenerate $2$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$. Note that, if $v$ can be lifted to a $\lambda $-cocartesian morphism $v: Y \rightarrow Z$ of $\operatorname{\mathcal{C}}$, then assumption $(1)$ guarantees that we can lift $\overline{\sigma }$ to a diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^-{ v } & \\ X \ar [ur]^{ u } \ar [rr]^{ w } & & Z } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$, where $w$ is $\lambda $-cocartesian by virtue of Proposition 5.1.4.12. Consequently, to prove the existence of $w$, we can replace $\overline{w}$ by $\overline{v}$ and thereby reduce to the case where $\lambda _{-}( \overline{w} )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$. Repeating this argument with the roles of $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ interchanged, we may also assume that $\lambda _{+}( \overline{w} )$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{+}$. In this case, assumption $(1)$ guarantees that we can lift $\overline{w}$ to an isomorphism $w: X \rightarrow Z$ in the $\infty $-category $\operatorname{\mathcal{C}}$, which is $\lambda $-cocartesian by virtue of Proposition 5.1.1.8. This completes the proof that $\lambda $ is a cocartesian fibration.

To complete the proof that $\lambda $ is a left fibration, it will suffice to show that every morphism $w: X \rightarrow Z$ in $\operatorname{\mathcal{C}}$ is $\lambda $-cocartesian (see Proposition 5.1.4.14). Arguing as in Remark 5.1.3.8, we can choose a diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^-{v} & \\ X \ar [ur]^{u} \ar [rr]^{w} & & Z } \]

in $\operatorname{\mathcal{C}}$, where $u$ is $\lambda $-cocartesian and $\lambda (v)$ is an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-} \times \operatorname{\mathcal{C}}_{+}$. It follows from $(2)$ that $v$ is $\lambda _{-}$-cocartesian. Since $\lambda _{-}(v)$ is an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}_{-}$, it follows that $v$ is an isomorphism in $\operatorname{\mathcal{C}}$ (Proposition 5.1.1.8). In particular, $v$ is $\lambda $-cocartesian (Proposition 5.1.1.8), so that $w = v \circ u$ is also $\lambda $-cocartesian (Proposition 5.1.4.12). $\square$

Proposition 8.2.1.8. Let $\lambda = (\lambda _{-}, \lambda _{+}): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a corepresentable coupling and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_{+}$ be a cocartesian fibration of $\infty $-categories. Suppose that, for every object $Y \in \operatorname{\mathcal{C}}_{+}$, the fiber $\{ Y \} \times _{ \operatorname{\mathcal{C}}_{+} } \operatorname{\mathcal{E}}$ has an initial object. Then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ has an initial object. Moreover, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is initial if and only if it satisfies the following pair of conditions:

$(1)$

For every couniversal object $C \in \operatorname{\mathcal{C}}$, the image $F(C)$ is initial when viewed as an object of the $\infty $-category $\{ \lambda _{+}(C) \} \times _{ \operatorname{\mathcal{C}}_{+} } \operatorname{\mathcal{E}}$.

$(2)$

The functor $F$ carries $\lambda _{+}$-cocartesian morphisms of $\operatorname{\mathcal{C}}$ to $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$.

Remark 8.2.1.9. By virtue of Proposition 8.2.1.7, we can restate condition $(2)$ of Proposition 8.2.1.8 as follows:

$(2')$

Let $e$ be a morphism of $\operatorname{\mathcal{C}}$ having the property that $\lambda _{-}(e)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$. Then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

Proof of Proposition 8.2.1.8. Set $\overline{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}_{+} } \operatorname{\mathcal{E}}$, and let $\overline{U}: \overline{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. Let $\operatorname{\mathcal{D}}$ denote the simplicial set $\operatorname{Res}_{\operatorname{\mathcal{C}}/\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( \overline{\operatorname{\mathcal{C}}} )$ of Construction 4.5.9.1, and let $\pi : \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ denote the projection map. By construction, we can identify $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ with the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{D}})$ of sections of $\pi $.

We first claim that $\pi $ is a cartesian fibration of $\infty $-categories. By virtue of Proposition 5.3.6.6, it will suffice to show that if $\overline{C}$ is an object of the $\infty $-category $\overline{\operatorname{\mathcal{C}}}$ and $s: C' \rightarrow \overline{U}( \overline{C} )$ is a $\lambda _{-}$-cocartesian morphism of $\operatorname{\mathcal{C}}$, then $s$ can be lifted to an $\overline{U}$-cartesian morphism $\overline{C}' \rightarrow \overline{C}$ of the $\infty $-category $\overline{\operatorname{\mathcal{C}}}$. Proposition 8.2.1.7 guarantees that $\lambda _{+}(s)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}_{+}$. Let $E$ denote the image of $\overline{C}$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Since the functor $U$ is an isofibration (Proposition 5.1.4.8), we can lift $\lambda _{+}(s)$ to an isomorphism $t: E' \rightarrow E$ in the $\infty $-category $\operatorname{\mathcal{E}}$. The pair $(s,t)$ can be regarded as a morphism $\overline{s}: \overline{C}' \rightarrow \overline{C}$ in the $\infty $-category $\overline{\operatorname{\mathcal{C}}}$. Since the morphism $t$ is $U$-cartesian (Proposition 5.1.1.8), the morphism $\overline{s}$ is $\overline{U}$-cartesian (Remark 5.1.1.11).

Fix an object $X \in \operatorname{\mathcal{C}}_{-}$, and set $\operatorname{\mathcal{C}}_{X} = \{ X\} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}} \operatorname{\mathcal{C}}$. Since $\lambda $ is corepresentable, the $\infty $-category $\operatorname{\mathcal{C}}_{X}$ has an initial object $C$. Set $Y = \lambda _{+}(C)$. By assumption, the $\infty $-category $\{ Y \} \times _{ \operatorname{\mathcal{C}}_{+} } \operatorname{\mathcal{E}}$ has an initial object. Invoking the criterion of Corollary 7.3.6.11, we see that the $\infty $-category

\[ \{ X\} \times _{ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\simeq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+} }( \operatorname{\mathcal{C}}_{X}, \operatorname{\mathcal{E}}) \]

also has an initial object. Moreover, an object $F_{X} \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+} }( \operatorname{\mathcal{C}}_{X}, \operatorname{\mathcal{E}})$ is initial if and only if it satisfies the following pair of conditions:

$(1_{X})$

For every initial object $C \in \operatorname{\mathcal{C}}_{X}$, the image $F_{X}(C)$ is an initial object of the $\infty $-category $\{ Y\} \times _{\operatorname{\mathcal{C}}_{+}} \operatorname{\mathcal{E}}$.

$(2_{X})$

The functor $F_{X}$ carries each morphism in $\operatorname{\mathcal{C}}_{X}$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

Applying Corollary 7.3.5.7, we deduce that the $\infty $-category

\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+} }( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \simeq \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{-}^{\operatorname{op}}}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}},\operatorname{\mathcal{D}}) \]

has an initial object. Moreover, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}_{+}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is initial if and only if, for every object $X \in \operatorname{\mathcal{C}}_{-}$, the restriction $F_{X} = F|_{ \operatorname{\mathcal{C}}_{X} }$ satisfies conditions $(1_{X} )$ and $(2_{X} )$ above. Unwinding the definitions, this is equivalent to the the requirement that $F$ satisfies condition $(1)$ and the following variant of condition $(2')$ of Remark 8.2.1.9:

$(2'')$

If $e$ is a morphism of $\operatorname{\mathcal{C}}$ such that $\lambda _{-}(e)$ is an identity morphism of $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$, then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

The implication $(2') \Rightarrow (2'')$ is immediate. The reverse implication follows from the observation that if $\lambda _{-}(e)$ is an isomorphism in $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$, then $e$ is isomorphic (as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$) to a morphism $e'$ such that $\lambda _{-}(e')$ is an identity morphism of $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$. $\square$

Corollary 8.2.1.10. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \{ C\} $ has an initial object. Then the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ has an initial object. Moreover, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})$ is initial if and only if it satisfies the following pair of conditions:

$(1)$

For every object $C \in \operatorname{\mathcal{C}}$, the image $F( \operatorname{id}_ C )$ is an initial object of the $\infty $-category $\operatorname{\mathcal{E}}_ C$.

$(2)$

Let $e$ be a morphism of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ whose image in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is an isomorphism. Then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{E}}$.

Stated more informally, Corollary 8.2.1.10 asserts that the twisted arrow $\infty $-category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is universal among $\infty $-categories $\operatorname{\mathcal{E}}$ equipped with a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having the property that each fiber of $U$ has an initial object.