Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $E,F\to X$ be 2 holomorphic vector bundles and $D\hookrightarrow X$ be a smooth divisor. Denote by $\mathcal{O}_X(D)$ the line bundle associated to the divisor and let $s: \mathcal{O}_X\to \mathcal{O}_X(D)$ be a section, such that $s^{-1}(0) = D$. Assume also that we have an isomorphism $E|_D\cong F|_D$ of the restrictions to $D$.

Consider the following perfect complex $\mathcal{E}^\bullet$ on $X$: \begin{equation*} E\oplus F\xrightarrow{ \begin{pmatrix} \text{id}\otimes s& 0\\ 0&\text{id}\otimes s \end{pmatrix}} (E\oplus F)\otimes \mathcal{O}_X(D)\xrightarrow{ \begin{pmatrix} \rho_E&\rho_F \end{pmatrix} } (E\otimes \mathcal{O}_X(D))|_D\,, \end{equation*} where $\rho_{(-)}$ corresponds to restriction of sections to $D$, and we use the isomorphism $F|_D\cong E|_D$.

Now it is well known that there exists a locally free resolution of this complex on $X$. However, is it possible to write it down explicitly from what we know?

**Edit:** By a locally free resolution I mean a complex of vector bundles $L^\bullet$ with a quasi-isomorphism $L^\bullet\to \mathcal{E}^\bullet$. It is important to me that I have such a map, as I would like to use it to construct a differential operator.