Restriction of sheaves to tangent space Let $B$ be a connected scheme and $X\to B$ a family of elliptic curves whose zero section is denoted by $0:B\to X$. For simplicity, we assume that the base field is $\mathbb{C}$. For any $\mathbb{C}$-scheme $Z$, let us denote $\mathrm{Sh}_F(Z)$ for the category of lisse topological $F$-sheaves where $F$ is some commutative ring.
Now, let $T$ be the vertical tangent line at $0$ and $T^\times\subset T$ be the complement to the zero section. Set $U=X-0(B)$. Under this setting, I found the following statement written without any justification in some literature:

there is an exact faithful tensor functor
  $\mathrm{Sh}_F(U)\to\mathrm{Sh}_F(T^\times)$ such that if $\mathcal{G}\in\mathrm{Sh}_F(X)$, it maps to $p^*\mathcal{G}_0$ where $p:T^\times\to B$ and $\mathcal{G}_0=0^*\mathcal{G}$.

Could anyone give a construction of such functor? Any suggestion would be helpful. Thank you in advance.
 A: This might be similar to (the Betti version of) the tangential basepoint construction in Deligne's "Le Groupe Fondamental de la Droite Projective
Moins Trois Points" paper: Section 15-Théorie classique in https://publications.ias.edu/sites/default/files/61_LeGroupeFondamentalDroite.pdf
The main idea is that one would choose a local real analytic isomorphism between a neighbourhood of the 0 section in T and in X which is identity on their respective tangent spaces (This last point is very important. Note that the corresponding tangent spaces are both T so it makes sense to require this map to be the identity map). One then restricts a sheaf then pulls-back and then notices that in the tangent space punctured neighbourhood has the same fundamental group as the punctured total space so one can extend to the total space. 
The remaining point is to show that this is independent of the choice of local real analytic isomorphism (with identity differential), but I think the same real analytic blow-up argument as in Deligne's paper would work in this case (I have not checked the details of this). 
An important point is that the construction is not algebraic, but is motivic (in the suitable sense). 
