Let $X$ be a manifold, $Z$ is its closed submanifold; for any $\varepsilon>0$ denote by $Z_\varepsilon$ the set of points of $X$ that lie at distance $<\varepsilon$ from $X$.

Now, for a sheaf $S$ (of abelian groups) on $X$ and each $\varepsilon>0$ we can consider a sheaf $S_\varepsilon:U\mapsto S(U\cap Z_\varepsilon)$, and also the limit sheaf $S_Z=\injlim_{\varepsilon\to 0} S_\varepsilon$.

My question is: does there exist a 'sheaf-theoretic' description of $S_Z$ (without describing each of its sections separately, and without mentioning $Z_\varepsilon$!)? Let $i$ denote the embedding $Z\to X$, and $i_\varepsilon$ denote the embedding $Z_\varepsilon\to X$. Then I would like to study the functor $W$ being the limit of $i_{\varepsilon\ast}i_{\varepsilon}^\ast$. Moreover, for any sheaf $F/X$ we should have $Hom(WS,F)\cong Hom(i^\ast S,i^\ast F)$. Certainly, the problem would be solved if $i^\ast$ possessed a left adjoint; yet it probably does not. In particular, $W$ cannot factorize through $i_*$ since otherwise $S_Z$ would be zero on $X\setminus Z$ (and this is wrong already for a constant $S$).

Any hints (or references) for dealing with my $W$ would be very welcome! I wouldn't like to introduce any 'infinitesmall' topology here (yet comments in this direction could also be quite interesting). And certainly I would like to study an algebraic analogue of this picture.^)