Let $X$ be a topological space and $\mathcal{G}$ a sheaf on it. Let $Z$ be a subspace of $X$. If $U\subset Z$ is open (in $Z$), we define $\mathcal{G}\big|_Z$ as the sheafification of the semi-sheaf $U\mapsto \varinjlim_{V\supset U}\mathcal{G}(V)$ (where $V$ is open in $X$).
If $\varphi:\mathcal{G}\to\iota_*\mathcal{F}$ is a morphism of sheaves (where $\iota:Z\hookrightarrow X$ is the inclusion), show that there is a natural induced morphism of sheaves $\varphi':\mathcal{G}\big|_Z\to\mathcal{F}$.
For an open set $U$ of $Z$, I'm trying to define $\varphi'_U:\mathcal{G}\big|_Z(U)\to\mathcal{F}(U)$ from the map $\varphi_U$.
If $s\in\mathcal{G}\big|_Z(U)$, we should find some open $U'$ of $X$ with $U=U'\cap Z$ and some $f\in\mathcal{G}(U')$ in order to define $\varphi'_{U}(s)=\varphi_{U'}(f)$, but I don't know how to do that.
How can I "unrestrict" $s\in\mathcal{G}\big|_Z(U)$ into some $f\in\mathcal{G}(U')$