I'm finding myself a little confused about Koszul-Malgrange holomorphic structures in a certain context.

Suppose $M$ is a complex manifold, $N$ is a smooth manifold with a smooth complex vector bundle $V$ and bundle connection $\nabla$, and $f:M\to N$ is a smooth map. We form the pullback bundle $f^*V$ over $M$ and give it the pullback connection $f^*\nabla$. Let $P: f^*V \to V$ be the associated smooth projection map. Assuming $f^*\nabla^{0,1}\cdot f^*\nabla^{0,1}=0$, a theorem of Koszul-Malgrange says there is a holomorphic structure on $f^*V$ that turns it into a holomorphic vector bundle over $M$.

My question is about the following. Suppose we have a (possibly locally defined) holomorphic section $s$ of $f^*V$. In a suitable trivialization over some open set $\Omega$, so that $f^*V$ splits as $\Omega\times \mathbb{C}^n$, we can write $$s(z) = (z, h(z))$$ where $h:\Omega\to \mathbb{C}^n$ is holomorphic.

Now, consider the map $$P\circ s: \Omega \to V$$ I'm basically wondering if it makes sense to write $P\circ s =(f(z), h(z))$ for some choice of trivialization of $V$. The sort-of roadblock that I'm hitting is that the trivialization of the pullback we used should come from the Koszul-Malgrange theorem, rather than be pullback of a trivialization of $V$. So, I'm finding it hard to fiddle the definition of the pullback bundle.

I'm mainly interested in the case where $\dim M \leq \dim N$ and $f$ is regular everywhere. Then in my trivizalization over $\Omega$, the map $f_*$ should be an isomorphism of $f^*V|_{\Omega}\to V|_{f(\Omega)}$ that acts by $(z,v)\mapsto (f(z),v)$. Of course, this induces a trivialization over $f(\Omega)$. Then we can find an open set $U$ containing $f(\Omega)$ in which we have a smooth trivialization that extends our trivialization over $f(\Omega)$. I *think* that in this trivialization, $P\circ s$ should assume the desired form.