Definition 9.2.5.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories which admit finite limits. We say that $F$ is left exact if it preserves finite limits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
9.2.5 Exact Functors
We begin by recalling some terminology.
Example 9.2.5.2. The terminology of Definition 9.2.5.1 originates in homological algebra. Suppose that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are abelian categories (Definition
The functor $F$ is left exact: that is, it preserves finite limits.
The functor $F$ preserves equalizers.
The functor $F$ preserves kernels. That is, for every exact sequence
in the abelian category $\operatorname{\mathcal{C}}$, the image
is an exact sequence in the abelian category $\operatorname{\mathcal{D}}$.
) and that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an additive functor (that is, a functor which preserves direct sums). In this case, the following conditions are equivalent:
\[ 0 \rightarrow X \xrightarrow {u} Y \xrightarrow {v} Z \]
\[ 0 \rightarrow F(X) \xrightarrow {F(u)} F(Y) \xrightarrow { F(v) } F(Z) \]
The equivalence $(1) \Leftrightarrow (2)$ follows from Corollary 7.6.4.25 (since $F$ is assumed to be additive). The equivalence $(2) \Leftrightarrow (3)$ follows from the observation that, for every pair of morphisms $f_0,f_1: Y \rightrightarrows Z$, the coequalizer $\operatorname{Eq}(f_0, f_1)$ can be identified with the kernel of the morphism $(f_1 - f_0): Y \rightarrow Z$ (see Proposition ).
Remark 9.2.5.3. In the situation of Example 9.2.5.2, we can replace $(3)$ by the following a priori weaker condition:
For every short exact sequence
in the abelian category $\operatorname{\mathcal{C}}$, the sequence
is also exact.
\[ 0 \rightarrow X \xrightarrow {u} Y \xrightarrow {v} Z \rightarrow 0 \]
\[ 0 \rightarrow F(X) \xrightarrow {F(u)} F(Y) \xrightarrow {F(v)} F(Z) \]
The implication $(3) \Rightarrow (3')$ is immediate. Conversely, suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an additive functor which satisfies condition $(3')$, and suppose we are given an exact sequence $0 \rightarrow X \xrightarrow {u} Y \xrightarrow {v} Z$ in $\operatorname{\mathcal{C}}$; we wish to show that the sequence $0 \rightarrow F(X) \xrightarrow {F(u)} F(Y) \xrightarrow {F(v)} F(Z)$ is exact in $\operatorname{\mathcal{D}}$. Applying condition $(3')$ to the short exact sequence
\[ 0 \rightarrow X \xrightarrow {u} Y \xrightarrow {q} \operatorname{coker}(u) \rightarrow 0, \]
we deduce that $F(u): F(X) \rightarrow F(Y)$ is a monomorphism, whose image is the kernel of the map of the map $F( Y ) \rightarrow F( \operatorname{coker}(u) )$. To show that this agrees with the kernel of $F(v)$, it will suffice to show that $v$ induces a monomorphism $F(\operatorname{coker}(u)) \rightarrow F(Z)$ in the abelian category $\operatorname{\mathcal{D}}$. This follows by applying condition $(3')$ to the short exact sequence
\[ 0 \rightarrow \operatorname{coker}(u) \rightarrow Z \rightarrow \operatorname{coker}(v) \rightarrow 0. \]
Variant 9.2.5.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite colimits. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is right exact if it preserves finite colimits: that is, if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is left exact. When $F$ is an additive functor between abelian categories, this condition can be restated as follows:
For every exact sequence
in the abelian category $\operatorname{\mathcal{C}}$, the sequence
is exact in the abelian category $\operatorname{\mathcal{D}}$.
\[ X \xrightarrow {u} Y \xrightarrow {v} Z \rightarrow \]
\[ F(X) \xrightarrow {F(u)} F(Y) \xrightarrow {F(v)} F(Z) \rightarrow 0 \]
Moreover, it suffices to verify condition $(\ast )$ in the special case where $u$ is an monomorphism (Remark 9.2.5.3).
In this section, we study an $\infty $-categorical counterpart of Definition 9.2.5.1.
Definition 9.2.5.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. We say that $F$ is left exact if, for every left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is cofiltered, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also cofiltered.
Variant 9.2.5.6. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. We say that $F$ is right exact if, for every right fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is filtered, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also filtered. Equivalently, $F$ is right exact if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is left exact.
Notation 9.2.5.7. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. We let $\operatorname{Fun}^{\operatorname{lex}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the left exact functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, and we let $\operatorname{Fun}^{\operatorname{rex}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by the right exact functors $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.
Theorem 9.2.5.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
If the functor $F$ is right exact (in the sense of Definition 9.2.5.5), then it preserves finite colimits.
If the $\infty $-category $\operatorname{\mathcal{C}}$ admits finite colimits and $F$ preserves finite colimits, then $F$ is right exact.
We will give a proof of Theorem 9.2.5.8 at the end of this section.
Corollary 9.2.5.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories which admit finite limits. Then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left exact (in the sense of Definition 9.2.5.1) if and only if the functor of $\infty $-categories $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is left exact (in the sense of Definition 9.2.5.5).
It will be useful to consider an infinitary version of Definition 9.2.5.5.
Definition 9.2.5.10. Let $\kappa $ be a regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is $\kappa $-left exact if, for every left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is $\kappa $-cofiltered, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also $\kappa $-cofiltered (see Definition 9.1.1.4). We say that $F$ is $\kappa $-right exact if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is $\kappa $-left exact. We let $\operatorname{Fun}^{\kappa -\operatorname{lex}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors which are $\kappa $-left exact, and $\operatorname{Fun}^{\kappa -\operatorname{rex}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ the full subcategory spanned by those functors which are $\kappa $-right exact.
Remark 9.2.5.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $F$ is left exact (in the sense of Definition 9.2.5.5) if and only if it is $\aleph _0$-left exact (in the sense of Definition 9.2.5.10).
Remark 9.2.5.12 (Monotonicity). Let $\kappa $ be a regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a $\kappa $-left exact functor of $\infty $-categories. Then $F$ is also $\kappa '$-left exact, for any regular cardinal $\kappa ' \leq \kappa $ (see Remark 9.1.1.9).
Remark 9.2.5.13 (Composition). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $\kappa $ be a regular cardinal. If $F$ and $G$ are $\kappa $-left exact, then the composition $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is also $\kappa $-left exact.
Example 9.2.5.14 (Morita Equivalence). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Morita equivalence of $\infty $-categories (see Definition 8.5.6.1). Then $F$ is $\kappa $-left exact for every regular cardinal $\kappa $. To prove this, choose a left fibration $\widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, where the $\infty $-category $\widetilde{\operatorname{\mathcal{D}}}$ is $\kappa $-cofiltered; we wish to show that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also $\kappa $-cofiltered. This follows from Remark 9.1.1.13, since the projection map $\widetilde{F}: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$ is also a Morita equivalence (Corollary 8.5.6.15).
Theorem 9.2.5.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. For every regular cardinal $\kappa $, the following conditions are equivalent:
The functor $F$ is $\kappa $-left exact.
For every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is $\kappa $-cofiltered.
Proof. For each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{D}}_{D/}$ has an initial object (Proposition 4.6.7.22), and is therefore $\kappa $-cofiltered (Example 9.1.1.6 ). Consequently, if $F$ is $\kappa $-left exact, then the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is also $\kappa $-cofiltered. We now prove the converse. Assume that, for each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is $\kappa $-cofiltered. Let $U: \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$ be a left fibration, where $\widetilde{\operatorname{\mathcal{D}}}$ is $\kappa $-cofiltered; we wish to show that the fiber product $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$ is also $\kappa $-cofiltered. Applying Example 7.4.5.6, we deduce that $\widetilde{\operatorname{\mathcal{C}}}$ can be realized as the colimit of a diagram
\[ \mathscr {F}: \widetilde{\operatorname{\mathcal{D}}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}\quad \quad X \mapsto ( \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}). \]
For each object $X \in \widetilde{\operatorname{\mathcal{D}}}$, the $\infty $-category $\mathscr {F}(X)$ is equivalent to $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{ U(X)/ }$ (Corollary 4.6.4.20), and is therefore $\kappa $-cofiltered. Since $\widetilde{\operatorname{\mathcal{D}}}^{\operatorname{op}}$ is $\kappa $-filtered, Proposition 9.1.5.14 implies that $\widetilde{\operatorname{\mathcal{C}}}$ is also $\kappa $-cofiltered. $\square$
Remark 9.2.5.16 (Exactness and Flatness). Let $\kappa $ be a regular cardinal and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Using the classification of left fibrations (see Corollary 5.6.0.6), we see that the following conditions are equivalent:
The functor $F$ is $\kappa $-left exact (Definition 9.2.5.10).
For every uncountable regular cardinal $\lambda $ and every $\kappa $-flat functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ (Definition 9.2.4.7), the composite functor
is also $\kappa $-flat.
\[ \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\xrightarrow {G} \operatorname{\mathcal{S}}_{< \lambda } \]
Moreover, if the $\infty $-category $\operatorname{\mathcal{D}}$ is locally $\lambda _0$-small (for some uncountable regular cardinal $\lambda _0$), then it suffices to verify condition $(b)$ in the special case where $\lambda = \lambda _0$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is a corepresentable functor (Theorem 9.2.5.15).
Corollary 9.2.5.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a left exact functor of $\infty $-categories. Then $F$ is right cofinal.
Proof. For every object $D \in \operatorname{\mathcal{D}}$, Theorem 9.2.5.15 guarantees that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is cofiltered. In particular, $\widetilde{\operatorname{\mathcal{C}}}$ is weakly contractible (Proposition 9.1.1.15). Allowing the object $D$ to vary, we conclude that $F$ is right cofinal (Theorem 7.2.3.1). $\square$
Corollary 9.2.5.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories which admits a left adjoint. Then $F$ is $\kappa $-left exact for every regular cardinal $\kappa $.
Proof. For every object $D \in \operatorname{\mathcal{D}}$, our assumption guarantees the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ has an initial object (given by $F(D)$; see Corollary 6.2.6.2). In particular, $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is $\kappa $-cofiltered (Example 9.1.1.6). Allowing $D$ to vary, we conclude that $F$ is $\kappa $-left exact (Theorem 9.2.5.15). $\square$
For sufficiently large $\kappa $, we have the following converse of Corollary 9.2.5.18:
Corollary 9.2.5.19. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\kappa $ be a regular cardinal. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small, and that $\operatorname{\mathcal{C}}$ is idempotent complete. Then $F$ admits a left adjoint $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ if and only if it is $\kappa $-left exact.
Proof. Assume that $F$ is $\kappa $-left exact; we will show that it admits a left adjoint (the converse is a special case of Corollary 9.2.5.18). By virtue of Corollary 6.2.6.2, it will suffice to show that for each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ has an initial object. Our assumption that $F$ is $\kappa $-left exact guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is $\kappa $-cofiltered (Theorem 9.2.5.15), and our assumption that $\operatorname{\mathcal{C}}$ is idempotent complete guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is also idempotent complete (Corollary 8.5.4.24). It will therefore suffice to show that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is essentially $\kappa $-small (Proposition 9.1.8.10). The $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small by assumption, and the assumption that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small guarantees that projection map $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is an essentially $\kappa $-small left fibration (Corollary 5.6.7.7). The desired result now follows from Corollary 4.7.9.12. $\square$
Warning 9.2.5.20. In the formulation of Corollary 9.2.5.19, the assumption that $\operatorname{\mathcal{C}}$ is idempotent complete cannot be omitted. For example, suppose that $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor which exhibits $\operatorname{\mathcal{D}}$ as an idempotent completion of $\operatorname{\mathcal{D}}$. Then $F$ is $\kappa $-left exact for every regular cardinal $\kappa $ (Example 9.2.5.14). However, $F$ admits a left adjoint only if it is an equivalence of $\infty $-categories: that is, only if the $\infty $-category $\operatorname{\mathcal{C}}$ is idempotent complete.
Corollary 9.2.5.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $\kappa $ be a regular cardinal, and let $G: K \rightarrow \operatorname{\mathcal{C}}$ be a $\kappa $-small diagram. If $F$ is $\kappa $-left exact, then the induced map of slice $\infty $-categories $F_{/G}: \operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{D}}_{/(F \circ G)}$ is also $\kappa $-left exact.
Proof. By virtue of Theorem 9.2.5.15, it will suffice to show that for each object $\widetilde{D} \in \operatorname{\mathcal{D}}_{/ (F \circ G)}$, the $\infty $-category
\[ \operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{/G} \times _{ \operatorname{\mathcal{D}}_{ / F \circ G} } (\operatorname{\mathcal{D}}_{ / F \circ G})_{ \widetilde{D}/ } \]
is $\kappa $-cofiltered. Let $D$ denote the image of $\widetilde{D}$ in $\operatorname{\mathcal{D}}$, and set $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$. Then $\widetilde{D}$ determines a lift of $G$ to a diagram $\widetilde{G}: K \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, and $\operatorname{\mathcal{E}}$ can be identified with the slice $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}_{ / \widetilde{G} }$. By Proposition 9.1.1.14, it will suffice to show that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ is $\kappa $-cofiltered, which follows from our assumption that $F$ is $\kappa $-left exact (Theorem 9.2.5.15). $\square$
Theorem 9.2.5.8 is a special case of the following:
Theorem 9.2.5.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\kappa $ be a regular cardinal. Then:
If the functor $F$ is $\kappa $-left exact, then it preserves $\kappa $-small limits.
If the $\infty $-category $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits and $F$ preserves $\kappa $-small limits, then $F$ is $\kappa $-left exact.
Proof. We first prove $(1)$. Assume that $F$ is $\kappa $-left exact; we wish to show that it preserves $\kappa $-small limits. Equivalently, we wish to show that if $G: K \rightarrow \operatorname{\mathcal{C}}$ is a $\kappa $-small diagram, then the functor $F_{/G}: \operatorname{\mathcal{C}}_{/G} \rightarrow \operatorname{\mathcal{D}}_{ /(F \circ G)}$ preserves final objects. It follows from Corollary 9.2.5.21 that the functor $F_{/G}$ is also $\kappa $-left exact. In particular, it is right cofinal (Corollary 9.2.5.17), and therefore preserves final objects by virtue of Corollary 7.2.1.9.
We now prove $(2)$. Assume that $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits and that $F$ preserves $\kappa $-small limits; we wish to show that $F$ is $\kappa $-left exact. By virtue of Theorem 9.2.5.15, it will suffice to show that for every object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is $\kappa $-cofiltered. Fix a $\kappa $-small simplicial set $K$ and a diagram $\widetilde{G}: K \rightarrow \widetilde{\operatorname{\mathcal{C}}}$; we wish to show that $\widetilde{G}$ can be extended to the cone $K^{\triangleleft }$. Let us identify $\widetilde{G}$ with a pair $(G, D')$, where $G: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram in $\operatorname{\mathcal{C}}$ and $D'$ is an object of the $\infty $-category $\operatorname{\mathcal{D}}_{/ (F \circ G)}$ lying over $D$. Since $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits, we can extend $G$ to a limit diagram $\overline{G}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Our assumption that $F$ preserves $\kappa $-small limits guarantees that $F \circ \overline{G}$ is a limit diagram in $\operatorname{\mathcal{D}}$: that is, the restriction functor $\operatorname{\mathcal{D}}_{ / (F \circ \overline{G} )} \rightarrow \operatorname{\mathcal{D}}_{ / (F \circ G)}$ is a trivial Kan fibration. In particular, we can lift $D'$ to an object $\overline{D}' \in \operatorname{\mathcal{D}}_{ / (F \circ \overline{G} )}$. The pair $(\overline{G}, \overline{D}' )$ can then determines an extension of $\widetilde{G}$ to a diagram $K^{\triangleleft } \rightarrow \widetilde{\operatorname{\mathcal{C}}}$. $\square$
Corollary 9.2.5.23. Let $\kappa $ and $\lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ be a functor of $\infty $-categories. Each of the following conditions implies the next:
The functor $F$ is $\kappa $-left exact.
The functor $F$ is $\kappa $-flat.
The functor $F$ preserves all $\kappa $-small limits which exist in $\operatorname{\mathcal{C}}$.
If $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits, then all three conditions are equivalent.
Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 9.2.5.16 and the implication $(2) \Rightarrow (3)$ is Corollary 9.2.4.19. If $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits, then the implication $(3) \Rightarrow (1)$ is a special case of Theorem 9.2.5.22. $\square$
Warning 9.2.5.24. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor of $\infty $-categories. It follows from Corollary 9.2.5.23 that if $F$ is left exact, then it is flat. Moreover, the converse holds if $\operatorname{\mathcal{C}}$ admits finite limits. Beware that the converse does not hold in general. For example, let $R$ be an associative ring, let $\operatorname{\mathcal{E}}$ be (the nerve of) the category of free left $R$-modules of finite rank, and let be the functor represented by $R$. Then $F$ is a flat functor. Using Remark 9.2.5.16, we see that $F$ is left exact if and only if, for every set $S$, the functor is also flat. By virtue of Example 9.2.4.3, this is equivalent the requirement that $R^{S}$ is a flat left $R$-module. This condition is satisfied only when the ring $R$ is right coherent: that is, every finitely generated right ideal $I \subseteq R$ is finitely presented as a right $R$-module. See Theorem 2.1 of [chase].
\[ F: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set}) \subseteq \operatorname{\mathcal{S}}\quad \quad F(M) = \operatorname{Hom}_{R}(M,R) \]
\[ F^{S}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set}) \subseteq \operatorname{\mathcal{S}}\quad \quad F(M) = \operatorname{Hom}_{R}(M,R^{S}) \]
Corollary 9.2.5.25. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small and admits $\kappa $-small colimits. Then the covariant Yoneda embedding exhibits $\operatorname{Fun}^{\kappa -\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$.
\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\kappa -\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) \]
Proof. Combine Theorem 9.2.4.14 with Corollary 9.2.5.23. $\square$
Corollary 9.2.5.26. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category which admits $\kappa $-small colimits. Then the covariant Yoneda embedding exhibits $\operatorname{Fun}^{\kappa -\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}$.
\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\kappa -\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]
Proof. Apply Corollary 9.2.5.25 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal. $\square$
Corollary 9.2.5.27. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category which admits finite colimits. Then the covariant Yoneda embedding exhibits $\operatorname{Fun}^{\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$.
\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]
Proof. Apply Corollary 9.2.5.26 in the special case $\kappa = \aleph _0$. $\square$