Remark 19.9.6. The arguments proving Lemmas 19.9.1 and 19.9.2 work also for *exact categories*, see [Appendix A, Buhler] and [1.1.4, BBD]. We quickly review this here and we add more details if we ever need it in the Stacks project.

Let $\mathcal{A}$ be an additive category. A *kernel-cokernel* pair is a pair $(i, p)$ of morphisms of $\mathcal{A}$ with $i : A \to B$, $p : B \to C$ such that $i$ is the kernel of $p$ and $p$ is the cokernel of $i$. Given a set $\mathcal{E}$ of kernel-cokernel pairs we say $i : A \to B$ is an *admissible monomorphism* if $(i, p) \in \mathcal{E}$ for some morphism $p$. Similarly we say a morphism $p : B \to C$ is an *admissible epimorphism* if $(i, p) \in \mathcal{E}$ for some morphism $i$. The pair $(\mathcal{A}, \mathcal{E})$ is said to be an *exact category* if the following axioms hold

$\mathcal{E}$ is closed under isomorphisms of kernel-cokernel pairs,

for any object $A$ the morphism $1_ A$ is both an admissible epimorphism and an admissible monomorphism,

admissible monomorphisms are stable under composition,

admissible epimorphisms are stable under composition,

the push-out of an admissible monomorphism $i : A \to B$ via any morphism $A \to A'$ exist and the induced morphism $i' : A' \to B'$ is an admissible monomorphism, and

the base change of an admissible epimorphism $p : B \to C$ via any morphism $C' \to C$ exist and the induced morphism $p' : B' \to C'$ is an admissible epimorphism.

Given such a structure let $\mathcal{C} = (\mathcal{A}, \text{Cov})$ where coverings (i.e., elements of $\text{Cov}$) are given by admissible epimorphisms. The axioms listed above immediately imply that this is a site. Consider the functor

exactly as in Lemma 19.9.2. It turns out that this functor is fully faithful, exact, and reflects exactness. Moreover, any extension of objects in the essential image of $F$ is in the essential image of $F$.

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