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Projections to the second summands define a morphism from that complex to the complex $$ F \stackrel{s}\to F(D)\tag{*} $$ of locally free sheaves. The cone of this morphism is the complex $$ 0 \to E \stackrel{s}\to E(D) \stackrel{\rho_E}\to E(D)\vert_D \to 0 $$ which is acyclic. Therefore, $(*)$ is a locally free resolution of the original complex.