Let $X$ and $Y$ be complex manifolds, let

\begin{align*} F: X \to Y \end{align*}

be a holomorphic map, and let $F^*: \Omega^\bullet(Y;\mathbb{C}) \to \Omega^\bullet(X;\mathbb{C})$ denote the smooth pullback of complex-valued differential forms. For every smooth complex-valued differential form $\alpha \in \Omega^\bullet(Y;\mathbb{C})$, where $\partial$ and $\bar{\partial}$ are the Dolbeault operators associated to the type decomposition of forms on $X$ and $Y$, one has

\begin{align*} F^*(\partial \alpha) = \partial(F^*\alpha) \end{align*}

and

\begin{align*} F^*(\bar{\partial}\alpha) = \bar{\partial}(F^*\alpha). \end{align*}