For a holomorphic map $f : M \to N$ between complex manifolds show that if $\omega$ is a form of type $(p,q)$ on $N$, then $f^*\omega$ is a form of type $(p,q)$ on $M$.
I am wondering if the following is enough to establish this. Let $\omega$ be form of type $(p,q)$. Locally $$\omega = \sum_{I,J} \omega_{I,J} dz_I \wedge d\bar{z}_J.$$ We have the following: $$f^*\omega = \sum_{I,J} \omega_{I,J} \circ f d(z_I \circ f) \wedge d(\bar{z}_J \circ f)$$
and $d(z_I \circ f)$ is a $(p,0)$ form and $d(\bar{z}_J \circ f)$ a $(0,q)$ form so the wedge is a $(p,q)$ form?