Let $X$ be a paracompact complex manifold, and let
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\begin{align*}
0 \to E \xrightarrow{i} F \xrightarrow{\pi} G \to 0
\end{align*}
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be a short exact sequence of holomorphic vector bundles over $X$, where $i: E \to F$ and $\pi: F \to G$ are holomorphic vector bundle morphisms. Fix an integer $p_0 \geq 0$. For every integer $q \geq 0$, define
by applying $i$ and $\pi$ to the coefficient vector bundle factor of a bundle-valued form. Then these maps commute with the corresponding Dolbeault differentials, and they assemble into a short exact sequence of cochain complexes