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Proposition 24.21.4. Let $X$ be a scheme.
  1. The category $\textit{QCoh}(\mathcal{O}_X)$ is a Grothendieck abelian category. Consequently, $\textit{QCoh}(\mathcal{O}_X)$ has enough injectives and all limits.
  2. The inclusion functor $\textit{QCoh}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ has a right adjoint $$ Q\footnote{This functor is sometimes called the coherator.} : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{QCoh}(\mathcal{O}_X) $$ such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism.

Proof. Part (1) means $\textit{QCoh}(\mathcal{O}_X)$ (a) has all colimits, (b) filtered colimits are exact, and (c) has a generator, see Injectives, Section 18.14. By Schemes, Section 22.24 colimits in $\textit{QCoh}(\mathcal{O}_X)$ exist and agree with colimits in $\textit{Mod}(\mathcal{O}_X)$. By Modules, Lemma 16.3.2 filtered colimits are exact. Hence (a) and (b) hold. To construct a generator $U$, pick a cardinal $\kappa$ as in Lemma 24.21.3. Pick a collection $(\mathcal{F}_t)_{t \in T}$ of $\kappa$-generated quasi-coherent sheaves as in Lemma 24.21.2. Set $U = \bigoplus_{t \in T} \mathcal{F}_t$. Since every object of $\textit{QCoh}(\mathcal{O}_X)$ is a filtered colimit of $\kappa$-generated quasi-coherent modules, i.e., of objects isomorphic to $\mathcal{F}_t$, it is clear that $U$ is a generator. The assertions on limits and injectives hold in any Grothendieck abelian category, see Injectives, Theorem 18.15.6 and Lemma 18.17.2.

Proof of (2). To construct $Q$ we use the following general procedure. Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_X)$ we consider the functor $$ \textit{QCoh}(\mathcal{O}_X)^{opp} \longrightarrow \textit{Sets},\quad \mathcal{G} \longmapsto \mathop{\rm Hom}\nolimits_X(\mathcal{G}, \mathcal{F}) $$ This functor transforms colimits into limits, hence is representable, see Injectives, Lemma 18.17.1. Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$ and a functorial isomorphism $\mathop{\rm Hom}\nolimits_X(\mathcal{G}, \mathcal{F}) = \mathop{\rm Hom}\nolimits_X(\mathcal{G}, Q(\mathcal{F}))$ for $\mathcal{G}$ in $\textit{QCoh}(\mathcal{O}_X)$. By the Yoneda lemma (Categories, Lemma 4.3.5) the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is functorial in $\mathcal{F}$. By construction $Q$ is a right adjoint to the inclusion functor. The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact that the inclusion functor $\textit{QCoh}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$ is fully faithful. $\square$

\begin{proposition}
\label{proposition-coherator}
Let $X$ be a scheme.
\begin{enumerate}
\item The category $\textit{QCoh}(\mathcal{O}_X)$ is a Grothendieck
abelian category. Consequently, $\textit{QCoh}(\mathcal{O}_X)$
has enough injectives and all limits.
\item The inclusion functor
$\textit{QCoh}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$
has a right adjoint
$$
Q\footnote{This functor is sometimes called the {\it coherator}.} :
\textit{Mod}(\mathcal{O}_X)
\longrightarrow
\textit{QCoh}(\mathcal{O}_X)
$$
such that for every quasi-coherent sheaf $\mathcal{F}$ the adjunction mapping
$Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism.
\end{enumerate}
\end{proposition}

\begin{proof}
Part (1) means $\textit{QCoh}(\mathcal{O}_X)$ (a) has all colimits,
(b) filtered colimits are exact, and (c) has a generator, see
Injectives, Section \ref{injectives-section-grothendieck-conditions}.
By Schemes, Section \ref{schemes-section-quasi-coherent}
colimits in $\textit{QCoh}(\mathcal{O}_X)$ exist and agree
with colimits in $\textit{Mod}(\mathcal{O}_X)$. By
Modules, Lemma \ref{modules-lemma-limits-colimits}
filtered colimits are exact. Hence (a) and (b) hold.
To construct a generator $U$, pick a cardinal $\kappa$ as in
Lemma \ref{lemma-colimit-kappa}. Pick a collection
$(\mathcal{F}_t)_{t \in T}$ of $\kappa$-generated quasi-coherent sheaves as in
Lemma \ref{lemma-set-of-iso-classes}. Set
$U = \bigoplus_{t \in T} \mathcal{F}_t$. Since every object of
$\textit{QCoh}(\mathcal{O}_X)$ is a filtered colimit of $\kappa$-generated
quasi-coherent modules, i.e., of objects isomorphic to $\mathcal{F}_t$,
it is clear that $U$ is a generator.
The assertions on limits and injectives hold in any
Grothendieck abelian category, see
Injectives, Theorem
\ref{injectives-theorem-injective-embedding-grothendieck} and
Lemma \ref{injectives-lemma-grothendieck-products}.

\medskip\noindent
Proof of (2). To construct $Q$ we use the following general procedure.
Given an object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_X)$
we consider the functor
$$
\textit{QCoh}(\mathcal{O}_X)^{opp} \longrightarrow \textit{Sets},\quad
\mathcal{G} \longmapsto \Hom_X(\mathcal{G}, \mathcal{F})
$$
This functor transforms colimits into limits,
hence is representable, see
Injectives, Lemma \ref{injectives-lemma-grothendieck-brown}.
Thus there exists a quasi-coherent sheaf $Q(\mathcal{F})$
and a functorial isomorphism
$\Hom_X(\mathcal{G}, \mathcal{F}) = \Hom_X(\mathcal{G}, Q(\mathcal{F}))$
for $\mathcal{G}$ in $\textit{QCoh}(\mathcal{O}_X)$. By the Yoneda lemma
(Categories, Lemma \ref{categories-lemma-yoneda})
the construction $\mathcal{F} \leadsto Q(\mathcal{F})$ is
functorial in $\mathcal{F}$. By construction $Q$ is a right
adjoint to the inclusion functor.
The fact that $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism
when $\mathcal{F}$ is quasi-coherent is a formal consequence of the fact
that the inclusion functor
$\textit{QCoh}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_X)$
is fully faithful.
\end{proof}

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