Let's work over a category of varieties on some field. Let $\mathcal{F}$ be a presheaf on varieties that satisfies Zariski/etale sheaf conditions whenever restricted to the opens of a specific variety. Is the injective resolution constructed through the Godement resolution a complex of sheaves which is also a presheaf on varieties compatible with $\mathcal{F}$?
For this problem we just need to consider the first embedding in the injective resolution which is given by $\mathcal{F}(X)\hookrightarrow \prod_{x\in X}J(\mathcal{F}_x)$, where the product is over the closed points of the variety $X$ and $J$ is the functorial injective embedding in the category of Abelian groups. In order to show that the following:
$$I_1 := \prod_{x\in X}J(\mathcal{F}_x)$$ is presheaf over the category of varieties compatible with $\mathcal{F}$. There seems to be an obvious way to extend functoriality of $\mathcal{F}$ to $I_1$, for example let $X\rightarrow Y$ be a morphism of varieties in order to define a morphism $I_1(Y)\rightarrow I_1(X)$ for every closed point $y$ in $Y$ we can look at its inverse image which will be a closed subset of $X$, denoted by $Z$. Given an element in $\mathcal{F}_y$ which is defined in some nbhd of $y$ we can pullback it to some nbhd of $Z$ and then look at its image in:
$$\prod_{z\in Z}\mathcal{F}_z$$ where $z$ varies over the closed points of $Z$. Now since $J$ is functorial this should still work after taking $J$, now letting $y$ vary over all closed points of $Y$ should define morphism from $I_1(Y)$ to $I_1(X)$ compatible with morphism from $F(Y)$ to $F(X)$.
Is this process correct? if so is there a cleaner way of seeing this? (I seemingly can derive contradictions using this, that is the main reason I am asking the question)