Injective abelian sheaves are flasque

Let $j_U:U\hookrightarrow X$ and $j_V:V\hookrightarrow X$ be the inclusions and let ${\mathbb{Z}}_U,{\mathbb{Z}}_V$ be the sheaves associated to the constant presheaves with value $\mathbb{Z}$ on $U$ and on $V$ [F3]. Since $V$ is open, the restriction $(j_{V!}{\mathbb{Z}}_V){\mathrm{\mid }}_U=j_U^{-1}{j}_{V!}{\mathbb{Z}}_V$ has sections over an open $W\subseteq U$ equal to $({\mathbb{Z}}_V)(W\cap V)=({\mathbb{Z}}_V)(W)={\mathbb{Z}}_U(W)$, so restriction along $U\subseteq V$ identifies it with ${\mathbb{Z}}_U$; concretely its sections over $W$ are the locally constant $\mathbb{Z}$-valued functions on $W$ with closed support in $W$ [F4]. Let $\beta :{\mathbb{Z}}_U\to (j_{V!}{\mathbb{Z}}_V){\mathrm{\mid }}_U$ be that identification and let $\alpha :j_{U!}{\mathbb{Z}}_U\to j_{V!}{\mathbb{Z}}_V$ be the morphism whose adjoint transpose under the adjunction [F2] is $\beta$; on sections over an open $W$ the map ${\alpha }_W$ extends a section of $\mathbb{Z}$ over $W\cap U$ with closed support in $W$ by zero across $W\setminus U$.

Combining the adjunction bijection [F2] with the presheaf correspondence of [F3] gives, for every open $W\subseteq X$, a natural bijection ${\mathrm{Hom}}_X(j_{W!}{\mathbb{Z}}_W,\mathcal{I})\cong \mathcal{I}(W)$: a morphism $j_{W!}{\mathbb{Z}}_W\to \mathcal{I}$ corresponds to ${\mathbb{Z}}_W\to \mathcal{I}{\mathrm{\mid }}_W$, and by [F3] this corresponds to the image of $1\in \mathbb{Z}={\mathbb{Z}}_W(W)$ in $\mathcal{I}(W)$; a morphism of sheaves $\mathcal{I}\to {\mathcal{I}}^{\mathrm{\prime }}$ or an inclusion $W\subseteq W^{\mathrm{\prime }}$ acts by the corresponding map of section groups, so the bijection is natural and, for $W\subseteq W^{\mathrm{\prime }}$, it carries the element of $\mathcal{I}(W^{\mathrm{\prime }})$ to its restriction in $\mathcal{I}(W)$.

For every open $W\subseteq X$ the map ${\alpha }_W$ is injective, because it is the inclusion of the sections over $W\cap U$ into the sections over $W\cap V$ given by extension by zero [F4, step 1.1]. By [F5] the kernel sheaf $\ker (\alpha )$ is computed objectwise, so $\ker (\alpha )(W)=0$ for every open $W$; hence $\ker (\alpha )=0$, and since $\mathrm{A}\mathrm{b}(X)$ is abelian [F5] the morphism $\alpha$ is a monomorphism.

Let $s\in \mathcal{I}(U)$ and let $f_s:j_{U!}{\mathbb{Z}}_U\to \mathcal{I}$ correspond to $s$ under the bijection of step 1.2 applied to $U$. By step 2.1 the morphism $\alpha$ is a monomorphism, so [F1] applied to the subobject $\alpha :j_{U!}{\mathbb{Z}}_U\rightarrowtail j_{V!}{\mathbb{Z}}_V$ and to the morphism $f_s$ provides $g:j_{V!}{\mathbb{Z}}_V\to \mathcal{I}$ with $g\circ \alpha =f_s$. Let $t\in \mathcal{I}(V)$ correspond to $g$ under the bijection of step 1.2 applied to $V$. Precomposition with $\alpha$ corresponds, under the two bijections of step 1.2, to the identification ${\mathbb{Z}}_U\cong (j_{V!}{\mathbb{Z}}_V){\mathrm{\mid }}_U$ of step 1.1, so by the naturality of step 1.2 the element corresponding to $g\circ \alpha$ is the restriction $t{\mathrm{\mid }}_U$; the identity $g\circ \alpha =f_s$ therefore says $t{\mathrm{\mid }}_U=s$. Hence every $s\in \mathcal{I}(U)$ extends to $\mathcal{I}(V)$, that is, ${\rho }_U^V$ is surjective, and since $U\subseteq V$ were arbitrary open subsets, $\mathcal{I}$ is flasque [F6]. ∎ [F1, F2, F3, F6, step 2.1, step 1.2]