The Stacks project

Remark 103.10.5. Let $\mathcal{X}$ be an algebraic stack. The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is abelian, the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ is right exact, but not exact in general, see Sheaves on Stacks, Lemma 96.15.1. We can use the functor $Q$ from Lemmas 103.10.1 and 103.10.2 to understand this. Namely, let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. Then

1. the cokernel $\mathop{\mathrm{Coker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is quasi-coherent and is the cokernel of $\varphi$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$,

2. the image $\mathop{\mathrm{Im}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is quasi-coherent and is the image of $\varphi$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$, and

3. the kernel $\mathop{\mathrm{Ker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ by Proposition 103.8.1 and $Q(\mathop{\mathrm{Ker}}(\varphi ))$ is the kernel in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

This follows from the references given.