What can be said about this morphism of sheaves

Let $X$ be topological space and $\cal F$ be a sheaf of modules over a sheaf of rings $\mathcal{O}$. One can consider an skyscraper functor $S(x,-): {\cal O}_{X,x}-{\rm Mod} \longrightarrow Sh(T)$ which assings to each module $M$ a sheaf of modules $S(x,M)$. This sheaf is defined by $U \mapsto M$ if $x\in U$ and $0$ otherwise.

The skyscraper functor and the stalk functor are adjoint i.e. $${\rm Hom}({\cal F}_x,A) = {\rm Hom}({\cal F}, S(x,A)).$$

Considering ${\cal F}$ and taking the injective envelope $E({\cal F}_x)$ for each $x\in X$, we get some injections ${\cal F}_x\rightarrow E({\cal F}_x)$ which by the adjoint property we get a morphism of sheaves ${\cal F}\rightarrow S(x,E({\cal F}_x))$ for each point $x$ of $X$.

These morphisms are injection (aren't these?) because of being injection on stalks. So we can assume the following morphism $${\cal F}\stackrel{\phi}\longrightarrow \bigoplus_{x\in X}S(x,E({\cal F}_x)).$$

What can be said about $\phi$? Is it an injection map? If not, which conditions need to impose to deduce injectivity? What can be said in the case that $X$ is a scheme?

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Yes, this is how you show that categories of sheaves/$\mathcal{O}_X$-modules have enough injectives, see for example the first few sections of Chapter III of Hartshorne. –  ChrisLazda May 7 '13 at 7:48
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