Exterior algebra decomposition natural projections $ Π^{p,q}:\bigwedge {V}^*_{\mathbb{C}} \to \bigwedge^{p,q}V$ (Huybrechts book)

Question

Let $\bigwedge^{p,q}V:=\bigwedge^{p}V^{1,0}\wedge \bigwedge^{p}V^{0,1}$ and $\bigwedge {V}^*_{\mathbb{C}}=\bigoplus_k \bigwedge^{k} V_{\mathbb{C}} $.

Then one defines the natural projection $$ Π^{p,q}:\bigwedge {V}^*_{\mathbb{C}} \to \bigwedge^{p,q}V$$

Is the projection defined as: $$(v_1,w_1, v_1\wedge w_1, \ldots, v_1\wedge\ldots \wedge v_q \wedge w_1\ldots \wedge w_{p})\mapsto v_1\wedge\ldots \wedge v_q \wedge w_1\ldots \wedge w_{p} ?$$

Where $v_i\in V^{1,0}$ and $w_i\in V^{0,1}$. I also use that $\bigwedge^{k}V_{\mathbb{C}}=\bigoplus_{p+q=k}\bigwedge^{p,q}V.$