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7.1.5 Preservation of Limits and Colimits

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Beware that, in general, $F$ need not carry (co)limit diagrams in $\operatorname{\mathcal{C}}$ to (co)limit diagrams in $\operatorname{\mathcal{D}}$. This motivates the following:

Definition 7.1.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $K$ be a simplicial set. We will say that $F$ preserves $K$-indexed limits if, for every limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, the composite map $(F \circ \overline{q}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$. We will say that $F$ preserves $K$-indexed colimits if, for every colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, the composite map $(F \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Example 7.1.5.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be any functor of $\infty $-categories. Then $F$ preserves $\Delta ^{0}$-indexed limits and colimits. By virtue of Example 7.1.4.10, this is equivalent to the observation that $F$ carries isomorphisms in $\operatorname{\mathcal{C}}$ to isomorphisms in $\operatorname{\mathcal{D}}$ (see Remark 1.4.1.6).

Warning 7.1.5.3. In the formulation of Definition 7.1.5.1, it is not necessary to assume that the $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits or colimits. For example, if $\operatorname{\mathcal{C}}$ is an $\infty $-category which contains no limit diagrams $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, then every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed limits. In practice, we will usually (but not always) apply the terminology of Definition 7.1.5.1 in cases where the $\infty $-category admits $K$-indexed limits or colimits, so that the conclusion of Definition 7.1.5.1 is non-vacuous.

Exercise 7.1.5.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $K$ be a simplicial set. Show that $F$ preserves $K$-indexed limits if and only if it satisfies the following condition:

  • For every diagram $u: K \rightarrow \operatorname{\mathcal{C}}$ and every natural transformation $\alpha : \underline{Y} \rightarrow u$ which exhibits an object $Y \in \operatorname{\mathcal{C}}$ as a limit of $u$ (in the sense of Definition 7.1.1.1), the image $F(\alpha ): \underline{F(Y)} \rightarrow (F \circ u)$ exhibits the object $F(Y) \in \operatorname{\mathcal{D}}$ as a limit of the diagram $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$.

Let us begin with a trivial example.

Proposition 7.1.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $K$ be a simplicial set. Then:

$(1)$

A morphism $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if the composition $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

$(2)$

A morphism $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if the composition $F \circ \overline{u}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

In particular, the equivalence $F$ preserves $K$-indexed limits and colimits.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram and set $u = \overline{u}|_{K}$. We then have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{ / \overline{u} } \ar [r] \ar [d] & \operatorname{\mathcal{D}}_{ / (F \circ \overline{u} )} \ar [d] \\ \operatorname{\mathcal{C}}_{/u} \ar [r] & \operatorname{\mathcal{D}}_{ / (F \circ \overline{u} )}. } \]

Since $F$ is an equivalence of $\infty $-categories, the horizontal maps in this diagram are also equivalences of $\infty $-categories (Corollary 4.6.4.18). It follows that the left vertical map is an equivalence of $\infty $-categories if and only if the right vertical map is an equivalence of $\infty $-categories. The desired result now follows from the criterion of Proposition 7.1.4.12. $\square$

Variant 7.1.5.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories and let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. If $F \circ \overline{u}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$, then $\overline{u}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$.

Corollary 7.1.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K$ be a simplicial set. Then:

$(1)$

Let $\overline{u}, \overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a limit diagram if and only if $\overline{v}$ is a limit diagram.

$(2)$

Let $\overline{u}, \overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams which are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$. Then $\overline{u}$ is a colimit diagram if and only if $\overline{v}$ is a colimit diagram.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $e: \overline{u} \rightarrow \overline{v}$ be an isomorphism in the $\infty $-category $\operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}})$. Under the canonical isomorphism

\[ \operatorname{Fun}( \Delta ^1, \operatorname{Fun}( K^{\triangleleft },\operatorname{\mathcal{C}}) ) \simeq \operatorname{Fun}( \Delta ^1 \times K^{\triangleleft }, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( K^{\triangleleft }, \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) ) \]

we can identify $e$ with a diagram $\overline{w}: K^{\triangleleft } \rightarrow \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$, which factors through the full subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Note that the evaluation maps

\[ \operatorname{ev}_0: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}} \]

are trivial Kan fibrations (Corollary 4.4.5.10), and therefore equivalences of $\infty $-categories (Proposition 4.5.2.10). By construction, we have $\overline{u} = \operatorname{ev}_0 \circ \overline{w}$ and $\overline{v} = \operatorname{ev}_1 \circ \overline{w}$. Applying Proposition 7.1.5.5, we deduce that $\overline{u}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $\overline{w}$ is a limit diagram in $\operatorname{Isom}(\operatorname{\mathcal{C}})$. Similarly, $\overline{v}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $\overline{w}$ is a limit diagram in $\operatorname{Isom}(\operatorname{\mathcal{C}})$. $\square$

Corollary 7.1.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given a pair of morphisms $u,v: K \rightarrow \operatorname{\mathcal{C}}$ which are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Then:

$(1)$

The morphism $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ if and only if $v$ can be extended to a limit diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.

$(2)$

The morphism $u$ can be extended to a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if $v$ can be extended to a colimit diagram $\overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Suppose that $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Since the diagrams $u$ and $v$ are isomorphic, it follows from Corollary 4.4.5.3 that $\overline{u}$ is isomorphic to a diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{v}|_{K} = v$. Applying Corollary 7.1.5.7, we conclude that $\overline{v}$ is also a limit diagram. $\square$

Corollary 7.1.5.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then:

$(1)$

The morphism $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ if and only if the composite map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ can be extended to a limit diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$.

$(2)$

The morphism $u$ can be extended to a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if the composite map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ can be extended to a colimit diagram $K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. If $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, then Proposition 7.1.5.5 guarantees that $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$ extending $F \circ u$. Conversely, suppose that $F \circ u$ can be extended to a limit diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an equivalence of $\infty $-categories which is homotopy inverse to $F$, so that $G \circ F$ is isomorphic to the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. Then $(G \circ \overline{v}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ (Proposition 7.1.5.5), and the restriction $(G \circ \overline{v})|_{K} = (G \circ F \circ u)$ is isomorphic to $u$ as an object of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Applying Corollary 7.1.5.8, we deduce that $u$ can be extended to a limit diagram $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. $\square$

Corollary 7.1.5.10. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories which are equivalent to one another, and let $K$ be a simplicial set. Then $\operatorname{\mathcal{C}}$ admits $K$-indexed (co)limits if and only if $\operatorname{\mathcal{D}}$ admits $K$-indexed (co)limits.

Remark 7.1.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $K$ be a simplicial set, and let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a limit diagram with restriction $u = \overline{u}|_{K}$. The following conditions are equivalent:

$(1)$

The composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.

$(2)$

For every limit diagram $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{u}'|_{K} = u$, the composition $(F \circ \overline{u}'): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.

The implication $(2) \Rightarrow (1)$ is immediate. For the converse, we observe that if $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is another limit diagram with $\overline{u}'|_{K} = u$, then $\overline{u}$ and $\overline{u}'$ are isomorphic when viewed as objects of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$, so that $F \circ \overline{u}$ and $F \circ \overline{u}'$ are isomorphic when viewed as objects of the $\infty $-category $\operatorname{\mathcal{D}}_{/ (F \circ u)}$. Since $F \circ \overline{u}$ is a final object of $\operatorname{\mathcal{D}}_{/ (F \circ u)}$, it follows that $F \circ \overline{u}'$ is also a final object of $\operatorname{\mathcal{D}}_{ / (F \circ u)}$ (Corollary 7.1.2.17).

A conservative functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which preserves $K$-indexed limits also reflects them:

Proposition 7.1.5.12. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $K$ be a simplicial set.

  • Suppose that $\operatorname{\mathcal{C}}$ admits $K$-indexed limits and the functor $F$ preserves $K$-indexed limits. Then a morphism $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

  • Suppose that $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits and the functor $F$ preserves $K$-indexed colimits. Then a morphism $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if $(F \circ \overline{u}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Proposition 7.1.5.12 is an immediate consequence of the following more precise assertion:

Lemma 7.1.5.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is also a limit diagram. Let $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be an arbitrary extension of $u$. Then $\overline{u}'$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if $F \circ \overline{u}'$ is a limit diagram in $\operatorname{\mathcal{D}}$.

Proof. Let us identify $\overline{u}$ and $\overline{u}'$ with objects $C$ and $C'$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$. Our assumption that $\overline{u}$ is a limit diagram guarantees that $C$ is a final object of $\operatorname{\mathcal{C}}_{/u}$, so there exists a morphism $f: C' \rightarrow C$ in $\operatorname{\mathcal{C}}_{/u}$. Note that $\overline{u}'$ is a limit diagram if and only if the object $C'$ is also final: that is, if and only if the morphism $f$ is an isomorphism.

Let $g: D' \rightarrow D$ be the image of $f$ under the functor $F_{/u}: \operatorname{\mathcal{C}}_{/u} \rightarrow \operatorname{\mathcal{D}}_{ / (F \circ u)}$. Our assumption that $F \circ \overline{u}$ is a limit diagram guarantees that $D$ is a final object of $\operatorname{\mathcal{D}}_{ / (F \circ u)}$. Consequently, $g$ is an isomorphism if and only if the object $D'$ is also final: that is, if and only if $(F \circ \overline{u}')$ is a limit diagram in $\operatorname{\mathcal{D}}$.

To complete the proof, it will suffice to show that $f$ is an isomorphism in $\operatorname{\mathcal{C}}_{/u}$ if and only if $g = F_{/u}(f)$ is an isomorphism in $\operatorname{\mathcal{D}}_{ / (F \circ q)}$. In fact, the functor $F_{/q}$ is conservative: this follows from our assumption that $F$ is conservative, by virtue of Corollary 4.4.2.12. $\square$

Definition 7.1.5.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $K$ be a simplicial set. We will say that the functor $F$ creates $K$-indexed limits if the following condition is satisfied:

  • Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ admits a limit in $\operatorname{\mathcal{D}}$. Then $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in $\operatorname{\mathcal{D}}$.

We say that the functor $F$ creates $K$-indexed colimits if it satisfies the following dual condition:

  • Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram for which the induced map $(F \circ u): K \rightarrow \operatorname{\mathcal{D}}$ admits a colimit in $\operatorname{\mathcal{D}}$. Then $u$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ for which the composition $(F \circ \overline{u}): K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram in $\operatorname{\mathcal{D}}$.

Remark 7.1.5.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $F$ creates $K$-indexed limits and that $F \circ u$ can be extended to a limit diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Then an extension $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $u$ is a limit diagram if and only if $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$ (see Lemma 7.1.5.13).

Proposition 7.1.5.16. Let $K$ be a simplicial set, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $K$-indexed limits, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories. The following conditions are equivalent:

$(1)$

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits and the functor $F$ preserves $K$-indexed limits.

$(2)$

The functor $F$ creates $K$-indexed limits.

Proof. The implication $(1) \Rightarrow (2)$ is immediately. Conversely, suppose that $(2)$ is satisfied and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Since $\operatorname{\mathcal{D}}$ admits $K$-indexed limits, $F \circ u$ can be extended to a limit diagram in $\operatorname{\mathcal{D}}$. Since $F$ creates $K$-indexed limits, it follows that there exists a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{u}|_{K} = u$ such that $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Applying Remark 7.1.5.11, we see that this holds for every limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{u}|_{K} =u$, which proves $(1)$. $\square$

The following is an important example of Definition 7.1.5.14:

Proposition 7.1.5.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $A$ be a simplicial set, and let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then:

$(1)$

The projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ creates $K$-indexed limits, for every simplicial set $K$.

$(2)$

The projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ creates $K$-indexed colimits, for every simplicial set $K$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Let $K$ be a simplicial set and let $p: K \rightarrow \operatorname{\mathcal{C}}_{f/}$ be a diagram, which we will identify with a morphism of simplicial sets $q: A \star K \rightarrow \operatorname{\mathcal{C}}$ satisfying $q|_{A} = f$. Set $g = q|_{K}$, so that $q$ can also be identified with a diagram $f': A \rightarrow \operatorname{\mathcal{C}}_{/g}$. Suppose that $g$ can be extended to a limit diagram $\overline{g}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Then the projection map $\operatorname{\mathcal{C}}_{/ \overline{g} } \rightarrow \operatorname{\mathcal{C}}_{/g}$ is a trivial Kan fibration (Proposition 7.1.4.12), so that $f'$ can be lifted to a diagram $f'': A \rightarrow \operatorname{\mathcal{C}}_{ / \overline{g} }$. We can then identify $f''$ with a morphism of simplicial sets $\overline{q}: A \star K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ extending $q$, or equivalently with a morphism $\overline{p}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{f/}$ extending $p$. We will complete the proof by showing that $\overline{p}$ is a limit diagram. To prove this, it will suffice to show that $\overline{p}$ is final when regarded as an object of the slice $\infty $-category $(\operatorname{\mathcal{C}}_{f/})_{/p} \simeq (\operatorname{\mathcal{C}}_{/g})_{f'/}$. This follows from Proposition 7.1.2.14, since $\overline{g}$ is a final object of $\operatorname{\mathcal{C}}_{/g}$. $\square$

Corollary 7.1.5.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: A \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $K$ be an arbitrary simplicial set. Then:

$(1)$

If $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, then the coslice $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ admits $K$-indexed limits and the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed limits.

$(2)$

If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, then the slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ admits $K$-indexed colimits and the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed colimits.

Corollary 7.1.5.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, so that $F$ induces a functor of coslice $\infty $-categories $F': \operatorname{\mathcal{C}}_{u/} \rightarrow \operatorname{\mathcal{D}}_{ (F \circ u)/}$. If the functor $F$ admits a right adjoint, then the functor $F'$ also admits a right adjoint.

Proof. We will show that $F'$ satisfies the criterion of Corollary 7.1.3.13. Fix an object $\overline{Y} \in \operatorname{\mathcal{D}}_{ (F \circ u) / }$, which we identify with a morphism of simplicial sets $K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ carrying the cone point of $K^{\triangleright }$ to an object $Y \in \operatorname{\mathcal{D}}$. We wish to show that the projection map

\[ U: \operatorname{\mathcal{C}}_{u/} \times _{ \operatorname{\mathcal{D}}_{ (F \circ u)/} } ( \operatorname{\mathcal{D}}_{ (F \circ u)/ } )_{/\overline{Y} } \rightarrow \operatorname{\mathcal{C}}_{u/} \]

is a representable right fibration. Unwinding the definitions, we can identify $\overline{Y}$ with a morphism of simplicial sets $\widetilde{u}: K \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y}$, and $U$ with the map $( \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )_{\widetilde{u} / } \rightarrow \operatorname{\mathcal{C}}_{u/}$. It will therefore suffice to show that the coslice $\infty $-category $(\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )_{ \widetilde{u} / }$ admits a final object. Since $F$ admits a right adjoint, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y}$ admits a final object (Corollary 7.1.3.13). The desired result now follows from the observation that the projection map

\[ (\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} )_{ \widetilde{u} / } \rightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/Y} \]

creates limits (Corollary 7.1.5.18). $\square$

Corollary 7.1.5.20. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. For every simplicial set $K$, the functor $F$ preserves $K$-indexed colimits and the functor $G$ preserves $K$-indexed limits.

Proof. We will show that $F$ preserves $K$-indexed colimits; the assertion that $G$ preserves $K$-indexed limits can be proved by a similar argument. Let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, so that $F$ induces a functor $F': \operatorname{\mathcal{C}}_{u/} \rightarrow \operatorname{\mathcal{D}}_{(F \circ u) / }$. We wish to show that the functor $F'$ carries initial objects of $\operatorname{\mathcal{C}}_{u/}$ to initial objects of $\operatorname{\mathcal{D}}_{ (F \circ u)/}$. It follows from Corollary 7.1.5.19 that the functor $F'$ also admits a right adjoint. We may therefore replace $F$ by $F'$ and thereby reduce to the case where $K = \emptyset $. In this case, we must show that if $X$ is an initial object of $\operatorname{\mathcal{C}}$, then $F(X)$ is an initial object of $\operatorname{\mathcal{D}}$. Choose an object $Y \in \operatorname{\mathcal{D}}$; we wish to show that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F( X ), Y )$ is a contractible Kan complex. Proposition 6.2.1.17 supplies a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X, G(Y) )$. We conclude by observing that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, G(Y) )$ is contractible, by virtue of our assumption that the object $X \in \operatorname{\mathcal{C}}$ is initial. $\square$