Let $f\colon X\to Y$ be a morphism of complex analytic spaces (though I'm very happy to restrict to complex manifolds).

Theorem (Grauert). The pushforward $f_*\colon\mathcal{O}_X\text{-mod}\to\mathcal{O}_Y\text{-mod}$ preserves coherence if $f$ is proper.

Question(s).

1. What conditions can we place on $f$ to ensure that the pullback $f^*\colon\mathcal{O}_Y\text{-mod}\to\mathcal{O}_X\text{-mod}$ preserves coherence?
2. Can we say anything about when such sufficient conditions are also necessary?
3. (Bonus: Can we also answer question 2 for the case of pushforward/proper?)

Let $f\colon X\to Y$ be a morphism of complex analytic spaces (though I'm very happy to restrict to complex manifolds).

Theorem (Grauert). The pushforward $f_*\colon\mathcal{O}_X\text{-mod}\to\mathcal{O}_Y\text{-mod}$ preserves coherence if $f$ is proper.

Question(s).

1. What conditions can we place on $f$ to ensure that the pullback $f^*\colon\mathcal{O}_Y\text{-mod}\to\mathcal{O}_X\text{-mod}$ preserves coherence?
2. Can we say anything about when such sufficient conditions are also necessary?
3. (Bonus: Can we also answer question 2 for the case of pushforward/proper?)

(I was sure that this question would already be on MO, and I did search but couldn't find anything)