Let $X$, $Y$, and $Z$ be complex manifolds, let $F:X \to Y$ be a holomorphic map, and let $p,q \geq 0$ be integers. The pullback of smooth differential forms by $F$ preserves forms of type $(p,q)$, commutes with the Dolbeault operator $\bar\partial$, and therefore induces a well-defined complex-[linear map](/page/Linear%20Map) $F^*:H^{p,q}_{\bar\partial}(Y) \to H^{p,q}_{\bar\partial}(X)$ on Dolbeault cohomology. Moreover, these induced maps are contravariantly functorial: for every holomorphic map $G:Y \to Z$, the induced maps satisfy $(G \circ F)^* = F^* \circ G^*:H^{p,q}_{\bar\partial}(Z) \to H^{p,q}_{\bar\partial}(X)$, and for the identity holomorphic map $\operatorname{id}_X:X \to X$, one has $\operatorname{id}_X^*=\operatorname{id}_{H^{p,q}_{\bar\partial}(X)}$.