Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative: $$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1_M \otimes_{\Omega^0_M} \mathcal{E} \xrightarrow{d^\nabla} \Omega^2_M \otimes_{\Omega^0_M} \mathcal{E} \xrightarrow{d^\nabla} \cdots$$ Here, I am using $\mathcal{E}$ to denote the sheaf of smooth sections of $E$, and $\Omega_M$ the sheaf of smooth differential forms on $M$.

We say that the connection $\nabla$ is flat if $(d^\nabla)^2 = 0$, and in this case we get an actual complex of sheaves. In this situation, the sheaf $\mathcal{L}$ of parallel sections is a local system and we can use this complex (a soft resolution) to compute its sheaf cohomology.

As described in Kashiwara and Schapira, or in Liviu Nicolaescu's answer here, given a closed submanifold $i: Z \hookrightarrow M$, and $j: M \setminus Z \hookrightarrow M$ the inclusion of its complement, we get a short exact sequence of sheaves $$0 \to j_!j^{-1}\mathcal{L} \to \mathcal{L} \to i_*i^{-1}\mathcal{L} \to 0.$$ The natural maps here are the counit $j_!j^{-1}\mathcal{L} \to \mathcal{L}$ of the $j_! \dashv j^{-1}$ adjunction, and the unit $\mathcal{L} \to i_*i^{-1}\mathcal{L}$ of the $i^{-1} \dashv i_*$ adjunction, respectively.

Note that the pullback bundle $i^* E \to Z$ inherits the flat connection from $E \to M$, and this gives rise to a complex of sheaves on $Z$ $$\mathcal{i^*E} \xrightarrow{d^\nabla=\nabla} \Omega^1_Z \otimes_{\Omega^0_Z} i^*\mathcal{E} \xrightarrow{d^\nabla} \Omega^2_Z \otimes_{\Omega^0_Z} i^*\mathcal{E} \xrightarrow{d^\nabla} \cdots$$ Note that $i^*= \Omega^0_Z \otimes_{i^{-1} \Omega^0_M} i^{-1}$, so it is not the same as the inverse image functor above. Is it correct to say that this complex gives a soft resolution of the sheaf $i^{-1} \mathcal{L}$? If so, how can I show that this is true?

This problem has come up in my research (in differential geometry) but my department doesn't really have anyone who works with sheaves. I would really appreciate some help with this!