Let $X$ be a manifold, $i: Z\to X$ is a closed embedding.

For a sheaf $S$ (of abelian groups) on a manifold $X$ and each  $\varepsilon>0$ we  denote by $Z_\varepsilon$ the set of points of $X$ that lie at distance $<\varepsilon$ from $X$. Consider the sheaf $S_\varepsilon:U\mapsto S(U\cap Z_\varepsilon)$, and also the section-wise limit $S_Z=\injlim_{\varepsilon\to 0} S_\varepsilon$.

I would like to understand the functor $W: S\to S_Z$. Unfortunately, $S_Z$ is only an section-wise injective limit of sheaves; so it does not have to be a sheaf (in the 'usual' sense; note that the topological space $X$ is not noetherian). In particular,  the stalk of $S_Z$ at any point $x\notin Z$ is easily seen to be $0$ (since $S_Z(U)=0$ for any $U$ such that its closure is disjoint from $Z$). On the other hand, the section $S_Z(X\setminus Z)$ does not seem to be $0$ (for example, if $S$ is constant); note that $X\setminus Z$ is not compact! See here the answer of algori to the previous version of this question (yesterday I believed that one can call $S_Z$ a sheaf).

Any hints (or references) for dealing with my $W$ would be very welcome! Note that $W$ is the limit of $i_{\varepsilon\ast}i_{\varepsilon}^\ast$. Moreover, it seems that for a sheaf $F/X$ we 'almost have': $Hom(S_Z,F)\cong Hom(i^\ast S,i^\ast F)\cong Hom(S,i_\ast i^\ast F)$, i.e. $W$ is 'almost an adjoint to $i_\ast i^\ast$'.  Are there any ideas how to work with such 'weird adjoints'? Possibly, the situation becomes somewhat better when one passes to etale sheaves over algebraic varieties.