# Kerodon

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### 7.3.3 Relative Kan Extensions

For many applications, it will be convenient to work with a generalization of Definition 7.3.2.1. In what follows, we assume that the reader is familiar with the theory of relative (co)limit diagrams introduced in ยง7.1.5.

Definition 7.3.3.1 (Relative Kan Extensions). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. For each object $C \in \operatorname{\mathcal{C}}$, we will say that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$ if the composite map

$(\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/C})^{\triangleright } \xrightarrow {c} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a $U$-colimit diagram in the $\infty$-category $\operatorname{\mathcal{D}}$. We say that $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$ if the composite map

$(\operatorname{\mathcal{C}}^{0}_{C/})^{\triangleleft } \hookrightarrow (\operatorname{\mathcal{C}}_{C/})^{\triangleleft } \xrightarrow {c'} \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}$

is a $U$-limit diagram in $\operatorname{\mathcal{D}}$. Here $c$ and $c'$ denote the slice and coslice contraction morphisms of Construction 4.3.5.12. We say that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$. We say that $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ at every object $C \in \operatorname{\mathcal{C}}$.

Remark 7.3.3.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$ (in the sense of Definition 7.3.2.1) if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ (in the sense of Definition 7.3.3.1), where $U: \operatorname{\mathcal{D}}\rightarrow \Delta ^0$ is the projection map. Similarly, $F$ is right Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$. See Example 7.1.5.3.

Remark 7.3.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories. Consider the evaluation functor

$\operatorname{ev}: \operatorname{\mathcal{C}}\times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}\quad \quad (C,F) \mapsto F(C).$

For every object $C \in \operatorname{\mathcal{C}}$ and every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, the following conditions are equivalent:

$(a)$

The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$.

$(b)$

The evaluation functor $\operatorname{ev}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ at $(C,F)$.

To prove this, it will suffice to show that the inclusion map

$\operatorname{\mathcal{C}}^{0}_{/C} \times \{ \operatorname{id}_{F} \} \hookrightarrow \operatorname{\mathcal{C}}^{0}_{/C} \times \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{/F}$

is right cofinal (Corollary 7.2.2.2). This follows from Corollary 7.2.1.19, since the inclusion map $\{ \operatorname{id}_{F} \} \hookrightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{/F}$ is right cofinal (the identity morphism $\operatorname{id}_{F}$ is an isomorphism in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, and therefore final when regarded as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})_{/F}$ by virtue of Proposition 4.6.7.22).

Remark 7.3.3.4. In the situation of Definition 7.3.3.1, the morphism $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is $U^{\operatorname{op}}$-left Kan extended from $(\operatorname{\mathcal{C}}^0)^{\operatorname{op}}$.

Example 7.3.3.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories. If $U$ is fully faithful, then $F$ is $U$-left Kan extended and $U$-right Kan extended from any full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ (see Example 7.1.5.4).

Example 7.3.3.6. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-left Kan extended from the empty subcategory $\emptyset \subseteq \operatorname{\mathcal{C}}$ if and only if it carries each object of $\operatorname{\mathcal{C}}$ to a $U$-initial object of $\operatorname{\mathcal{D}}$. Similarly, $F$ is $U$-right Kan extended from the empty subcategory if and only if it carries each object of $\operatorname{\mathcal{C}}$ to a $U$-final object of $\operatorname{\mathcal{D}}$.

To verify the Kan extension conditions of Definition 7.3.3.1, it suffices to consider objects $C$ which do not belong to the full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$.

Proposition 7.3.3.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories. Let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory and let $C \in \operatorname{\mathcal{C}}$ be an object which is isomorphic to an object of $\operatorname{\mathcal{C}}^{0}$. Then $F$ is both $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ and $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $C$.

Proof. We will show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ at $C$; the analogous statement for the right Kan extension condition follows by a similar argument. Let $c: (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be the slice contraction morphism; we wish to show that the composition $(F \circ c): (\operatorname{\mathcal{C}}_{/C}^{0})^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram. Choose an object $C' \in \operatorname{\mathcal{C}}^{0}$ and an isomorphism $u: C' \rightarrow C$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Our assumption that $u$ is an isomorphism guarantees that it is final when viewed as an object of the slice $\infty$-category $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.7.22), and therefore also when viewed as an object of the $\infty$-category $\operatorname{\mathcal{C}}_{/C}^{0}$. The desired result now follows from Corollary 7.2.3.6, since $F(u)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$. $\square$

Example 7.3.3.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories. Then $F$ is $U$-left Kan extended and $U$-right Kan extended from the full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{C}}$.

Example 7.3.3.9. Let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, and set $F = \overline{F}|_{\operatorname{\mathcal{C}}}$. Then $F$ is $U$-left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ if and only if $\overline{F}$ is $U$-left Kan extended from the cone $(\operatorname{\mathcal{C}}^0)^{\triangleright } \subseteq \operatorname{\mathcal{C}}^{\triangleright }$. To prove this, it suffices (by virtue of Proposition 7.3.3.7) to show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at an object $C \in \operatorname{\mathcal{C}}$ if and only if $\overline{F}$ is $U$-left Kan extended from $( \operatorname{\mathcal{C}}^{0} )^{\triangleright }$ at $C$, which follows immediately from the definition.

Example 7.3.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty$-categories. It follows from Proposition 7.3.3.7 that a functor $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a $U$-colimit diagram (in the sense of Definition 7.1.5.1) if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}$.

Proposition 7.3.3.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty$-categories, and let $\operatorname{\mathcal{C}}^0 \subseteq \operatorname{\mathcal{C}}$ be a coreflective subcategory of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

$(2)$

Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ which exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$ (Definition 6.2.2.1). Then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.

$(3)$

Let $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a right adjoint to the inclusion. If $e$ is a morphism in $\operatorname{\mathcal{C}}$ and $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}_0$, then $F(e)$ is a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.

Proof. Let $Y$ be an object of $\operatorname{\mathcal{C}}$. By assumption, there exists an object $X \in \operatorname{\mathcal{C}}^{0}$ and a morphism $e: X \rightarrow Y$ which exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$. Then $e$ is final when viewed as an object of the $\infty$-category $\operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$. It follows that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at $Y$ if and only if $F(e )$ is $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$; in particular, this condition is independent of the choice of $e$. Allowing the object $Y$ to vary, we deduce the equivalence $(1) \Leftrightarrow (2)$.

Using Lemma 6.2.2.14, we can choose a functor $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{0}$ and a natural transformation $\epsilon : T \rightarrow \operatorname{id}_{\operatorname{\mathcal{C}}}$ which exhibits $T$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection functor, so that $T$ is right adjoint to the inclusion of $\operatorname{\mathcal{C}}^{0}$ into $\operatorname{\mathcal{C}}$ (Proposition 6.2.2.15). Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. If $e$ exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$, then $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}^{0}$, which shows immediately that $(3)$ implies $(2)$. Conversely, suppose that $(2)$ is satisfied and that $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}^{0}$. We then have a commutative diagram

$\xymatrix@C =50pt@R=50pt{ (F \circ T)(X) \ar [r]^-{ (F \circ T)(e) } \ar [d]^{ F(\epsilon _{X})} & (F \circ T)(Y) \ar [d]^{ F( \epsilon _ Y) } \\ F(X) \ar [r]^-{ F(e) } & F(Y) }$

in the $\infty$-category $\operatorname{\mathcal{D}}$, where the upper horizontal map is an isomorphism and the vertical maps are $U$-cocartesian. Using Corollary 5.1.2.4, we see that $F(e)$ is also $U$-cocartesian. $\square$

Corollary 7.3.3.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a coreflective subcategory. The following conditions are equivalent:

$(1)$

The functor $F$ is left Kan extended from $\operatorname{\mathcal{C}}^0$.

$(2)$

Let $e: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ which exhibits $X$ as a $\operatorname{\mathcal{C}}^{0}$-coreflection of $Y$ (Definition 6.2.2.1). Then $F(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

$(3)$

Let $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_0$ be a right adjoint to the inclusion. If $e$ is a morphism in $\operatorname{\mathcal{C}}$ and $T(e)$ is an isomorphism in $\operatorname{\mathcal{C}}_0$, then $F(e)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

Proof. Combine Proposition 7.3.3.11 with Example 5.1.1.4 (for a closely related statement, see Proposition 7.3.1.17). $\square$

Corollary 7.3.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty$-categories, and suppose that $\operatorname{\mathcal{C}}$ contains an initial object. The following conditions are equivalent:

$(1)$

The functor $F$ is $U$-left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$ spanned by the initial objects.

$(2)$

The functor $F$ carries every morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{D}}$.

Corollary 7.3.3.14 (Constant Diagrams). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category which contains an initial object, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. The following conditions are equivalent:

$(1)$

The functor $F$ is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{init}} \subseteq \operatorname{\mathcal{C}}$ spanned by the initial objects.

$(2)$

The functor $F$ carries each morphism in $\operatorname{\mathcal{C}}$ to an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$.

$(3)$

The functor $F$ is isomorphic to a constant functor.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 7.3.3.13 by taking $\operatorname{\mathcal{E}}= \Delta ^0$, and the implication $(3) \Rightarrow (2)$ is immediate. To prove the converse, we observe that condition $(2)$ guarantees that $F$ can be regarded as a morphism from $\operatorname{\mathcal{C}}$ to the Kan complex $\operatorname{\mathcal{D}}^{\simeq }$. Since $\operatorname{\mathcal{C}}$ has an initial object, it is weakly contractible (Corollary 4.6.7.25), so this morphism is automatically nullhomotopic (Remark 3.2.4.18). $\square$

We now record some basic stability properties enjoyed by the class of relative Kan extensions, which follow easily from the analogous stability properties of relative (co)limit diagrams.

Remark 7.3.3.15. Suppose we are given a commutative diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [r]^-{G} \ar [d]^{U} & \operatorname{\mathcal{D}}' \ar [d]^{U'} \\ \operatorname{\mathcal{E}}\ar [r] & \operatorname{\mathcal{E}}', }$

where the horizontal functors are equivalence of $\infty$-categories. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G \circ F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Remark 7.1.5.6). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G \circ F$ is $U'$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Remark 7.3.3.16. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Let $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor which is isomorphic to $U$ (as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})$). Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $V$-left Kan extended from $\operatorname{\mathcal{C}}^0$ (see Remark 7.1.5.7). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^0$ if and only if it is $V$-right Kan extended from $\operatorname{\mathcal{C}}^0$.

Remark 7.3.3.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which is isomorphic to $F$ (as an object of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$), and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Proposition 7.1.5.13). Similarly, $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $G$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proposition 7.3.3.18 (Change of Source). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a replete full subcategory. Let $G: \operatorname{\mathcal{B}}\rightarrow \operatorname{\mathcal{C}}$ be an equivalence of $\infty$-categories, and set $\operatorname{\mathcal{B}}^{0} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{B}}$. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if $F \circ G$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$.

Proof. Assume first that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$; we will show that $F \circ G$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$. Fix an object $B \in \operatorname{\mathcal{B}}$ and set $\operatorname{\mathcal{B}}^{0}_{/B} = \operatorname{\mathcal{B}}^{0} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{B}}_{/B}$; we wish to show that the composite map

$\theta : ( \operatorname{\mathcal{B}}^{0}_{/B} )^{\triangleright } \hookrightarrow \operatorname{\mathcal{B}}_{/B}^{\triangleright } \rightarrow \operatorname{\mathcal{B}}\xrightarrow { F \circ G} \operatorname{\mathcal{D}}$

is a $U$-colimit diagram. Set $C = G(B)$ and $\operatorname{\mathcal{C}}^0_{/C} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}$. Since $G$ is an equivalence of $\infty$-categories, the induced map $G_{/B}: \operatorname{\mathcal{B}}_{/B} \rightarrow \operatorname{\mathcal{C}}_{/C}$ is also an equivalence of $\infty$-categories (Corollary 4.6.4.19). Our assumption that $\operatorname{\mathcal{C}}^{0}$ is a replete subcategory of $\operatorname{\mathcal{C}}$ guarantees that $\operatorname{\mathcal{C}}_{/C}^{0}$ is a replete subcategory of $\operatorname{\mathcal{C}}_{/C}$. In particular, the inclusion map $\operatorname{\mathcal{C}}_{/C}^{0} \hookrightarrow \operatorname{\mathcal{C}}_{/C}$ is an isofibration, so that $G_{/B}$ restricts to an equivalence of $\infty$-categories $G_{/B}^{0}: \operatorname{\mathcal{B}}_{/B}^{0} \rightarrow \operatorname{\mathcal{C}}_{/C}^{0}$. By construction, the morphism $\theta$ is the composition of $(G_{/B}^{0})^{\triangleright }$ with the map

$\theta ': ( \operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}_{/C}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}},$

which is a $U$-colimit diagram by virtue of our assumption that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Applying Corollary 7.2.2.2, we deduce that $\theta$ is also a $U$-colimit diagram.

We now prove the converse. Assume that $F \circ G$ is $U$-left Kan extended from $\operatorname{\mathcal{B}}^{0}$; we wish to show that $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Let $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ be a homotopy inverse to $G$, so that $(G \circ H): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. Since $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ is replete, it coincides with the inverse image $(G \circ H)^{-1} \operatorname{\mathcal{C}}^{0} = H^{-1} \operatorname{\mathcal{B}}^{0}$. Applying the first part of the proof, we deduce that the functor $(F \circ G \circ H): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. The functor $F$ is isomorphic to $F \circ G \circ H$, and is therefore also $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (Remark 7.3.3.17). $\square$

Remark 7.3.3.19 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$, and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ be functors of $\infty$-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that $U \circ F$ is $V$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Then $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $(V \circ U)$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (see Proposition 7.1.5.14). Similarly, if $U \circ F$ is $V$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $(V \circ U)$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Remark 7.3.3.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty$-categories, and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that $U \circ F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$. Then $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$; this follows by applying Remark 7.3.3.19 in the special case $\operatorname{\mathcal{E}}' = \Delta ^0$. Similarly, if $U \circ F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is right Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if it is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proposition 7.3.3.21 (Base Change). Suppose we are given a commutative diagram of $\infty$-categories

7.14
$$\begin{gathered}\label{equation:base-change-relative-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [rr]^{H'} \ar [dr] \ar [dd]^{G} & & \operatorname{\mathcal{E}}' \ar [dl] \ar [dd] \\ & \operatorname{\mathcal{B}}' \ar [dd] & \\ \operatorname{\mathcal{D}}\ar [dr]_{ U } \ar [rr]^(.4){H} & & \operatorname{\mathcal{E}}\ar [dl]^{V} \\ & \operatorname{\mathcal{B}}& } \end{gathered}$$

where each square is a pullback and the diagonal maps are inner fibrations. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be a functor of $\infty$-categories and $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Then:

$(1)$

If $G \circ F$ is $H$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $H'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

$(2)$

Assume that $U$ and $V$ are cartesian fibrations and that the functor $G$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cartesian morphisms of $\operatorname{\mathcal{E}}$. If $F$ is $H'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $G \circ F$ is an $H$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$.

Proof. Use Proposition 7.1.5.19. $\square$

Corollary 7.3.3.22. Suppose we are given a pullback diagram of $\infty$-categories

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}' \ar [d]^{U'} \ar [r]^-{G} & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{E}}' \ar [r] & \operatorname{\mathcal{E}}, }$

where the vertical maps are inner fibrations. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}'$ be a functor of $\infty$-categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $G \circ F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$, then $F$ is $U'$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$. The converse holds if $U$ is a cartesian fibration.

Proof. Apply Proposition 7.3.3.21 in the special case $\operatorname{\mathcal{B}}= \operatorname{\mathcal{E}}$. $\square$

Corollary 7.3.3.23. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an inner fibration of $\infty$-categories, let $\operatorname{\mathcal{D}}_{E} = \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ be the fiber of $U$ over an object $E \in \operatorname{\mathcal{E}}$, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}_{E}$ be a functor of $\infty$-categories, and $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. If $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (when regarded as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$), then it is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ (when regarded as a functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}_{E}$). The converse holds if $U$ is a cartesian fibration.

Proof. Apply Corollary 7.3.3.22 in the special case $\operatorname{\mathcal{E}}' = \{ E\}$. $\square$

Remark 7.3.3.24. In the situation of Corollary 7.3.3.23, the functor $F$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if and only if, for every morphism $f: E \rightarrow E'$ in the $\infty$-category $\operatorname{\mathcal{E}}$, the composite map

$\operatorname{\mathcal{C}}\xrightarrow {F} \{ E\} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$

is left Kan extended from the full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. See Remark 7.1.5.23.