Morphisms of sheaves on a covering

Question

I have encountered the following situation: Suppose we are given a site C admitting a final object $X$, together with a covering $\tilde{X}\to X$ by a single object. Now I have a sheaf $\mathcal{F}$ on $X$, such that $\mathcal{F}|_{\tilde{X}}$ is a constant sheaf associated to an abelian group $A$, in symbols $\mathcal{F}|_{\tilde{X}}=\underline{A}$.

I would now like to understand morphisms $Hom(\mathcal{F},\mathcal{G})$ into some abelian sheaf $\mathcal{G}$. Now this is where my confusion is:

1. We may consider the slice category $C/\tilde{X}$ which is a base for the site C. Then, the restriction functor $\mathcal{G} \mapsto \mathcal{G}|_{\tilde{X}}$ induces an equivalence of categories between the categories of sheaves on $C$, and the categories of sheaves on $C/\tilde{X}$. With this reasoning, $Hom(\mathcal{F},\mathcal{G})=Hom(\mathcal{F}|_{\tilde{X}},\mathcal{G}|_{\tilde{X}})=Hom(\underline{A},\mathcal{G}|_{\tilde{X}})=Hom(A,\mathcal{G}(\tilde{X}))$.

2. On the other hand, we have the sheaf $\mathscr{H}\kern -.90pt om(\mathcal{F},\mathcal{G})$. Using this, we may compute $Hom(\mathcal{F},\mathcal{G}) =\mathscr{H}\kern -.90pt om(\mathcal{F},\mathcal{G})(X) =\ker\left(\mathscr{H}\kern -.90pt om(\mathcal{F},\mathcal{G})(\tilde{X})\to\mathscr{H}\kern -.90pt om(\mathcal{F},\mathcal{G})(\tilde{X}\times_{X}\tilde{X})\right) =\ker\left(Hom(\mathcal{F}|_{\tilde{X}},\mathcal{G}|_{\tilde{X}})(\tilde{X})\to Hom(\mathcal{F}|_{\tilde{X}\times_{X}\tilde{X}},\mathcal{G}|_{\tilde{X}\times_{X}\tilde{X}})\right) =\ker\left(Hom(\underline{A}|_{\tilde{X}},\mathcal{G}|_{\tilde{X}})\to Hom(\underline{A}|_{\tilde{X}\times_{X}\tilde{X}},\mathcal{G}|_{\tilde{X}\times_{X}\tilde{X}})\right) =\ker\left(Hom(A,\mathcal{G}(\tilde{X}))\to Hom(A,\mathcal{G}(\tilde{X}\times_{X}\tilde{X}))\right) =Hom(A,\ker\left(\mathcal{G}(\tilde{X}))\to \mathcal{G}(\tilde{X}\times_{X}\tilde{X}) \right)) =Hom(A,\mathcal{G}(X))$.

But in general $Hom(A,\mathcal{G}(\tilde{X}))\neq Hom(A,\mathcal{G}(X))$. Something must have gone wrong: Where is my mistake?

Remark: The specific situation I am thinking about is in Proposition 6.10 here: https://www.math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf