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 $W: S\to S_Z$ (I do not want to describe it section-wisely). Unfortunately, $S_Z$ is only an in-sheaf; 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 a 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 $i_{\varepsilon\ast}i_{\varepsilon}^\ast$. Unfortunately, the adjunctions that I wrote about previously probably do not hold (in any sense).

P.S. Some observations that do not seem to help me.

1. My definition of $S_Z$ extends (without any changes) to presheaves.

2. $W':P\mapsto P_Z$ sends ind-sheaves to ind-sheaves.

3. It seems that ind-sheaves are (exactly) sheaves for the Grothendieck topology that admits only 'finite' covers.

So, perhaps one should use (somehow) the interplay between sheaves, ind-sheaves, and presheaves (and the corresponding topologies).