Lemma 64.18.3. The category of $\mathbf{Z}_\ell $-sheaves on $X$ is abelian.

**Proof.**
Let $\Phi = \left\{ \varphi _ n\right\} _{n\geq 1} : \left\{ \mathcal{F}_ n\right\} \to \left\{ \mathcal{G}_ n\right\} $ be a morphism of $\mathbf{Z}_\ell $-sheaves. Set

and $\mathop{\mathrm{Ker}}(\Phi )$ is the result of Lemma 64.18.2 applied to the inverse system

That this defines an abelian category is left to the reader. $\square$

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