Remark 62.3.9 (Open embeddings and compactly supported sections). Let $X$ be a separated scheme locally of finite type over a field $k$. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Exactly as in Remark 62.3.5 for $X' \subset X$ open there is an injective map

$H^0_ c(X', \mathcal{F}|_{X'}) \longrightarrow H^0_ c(X, \mathcal{F})$

and these maps turn $H^0_ c$ into a “cosheaf” on the Zariski site of $X$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).