I'm having trouble understanding this 'natural' isomorphism when discussing complex differential forms. Let $(M, J)$ be an almost complex manifold. Then its complexified cotangent bundle decomposes as $(T^* M \otimes \mathbb{C}) = T^*M^{(1,0)} \oplus T^*M^{(0,1)}$. Therefore its exterior algebra also decomposes into $\Lambda^k (T^* M \otimes \mathbb{C}) = \bigoplus_{p+q=k} \Lambda^p(T^* M^{(1,0)}) \otimes \Lambda^q(T^* M^{(0,1)}) =: \bigoplus_{p+q=k} \Lambda^{(p,q)} (T^* M \otimes \mathbb{C})$. Let $v^I$ be a basis for $\Lambda^p(T^* M^{(1,0)})$ and $w^J$ basis for $\Lambda^q(T^* M^{(0,1)})$, then
$$ \eta = \sum_{|I| = p, |J| = q} a_{IJ} v^I \otimes w^J \in \Lambda^{(p,q)}(T^* M \otimes \mathbb{C}). $$
This is all (somewhat) understandable. I have trouble with the following identification;
$$ \eta \mapsto \sum_{|I| = p, |J| = q} a_{IJ} v^I \wedge w^J \in \Lambda^{p+q}(T^* M \otimes \mathbb{C}), $$
is supposed to be a canonical isomorphism (onto its image). To me it looks like we just applied $Alt$ to $\eta$. In what way is this an isomorphism, or why can we identify element of $\Lambda^{(p,q)}(T^* M \otimes \mathbb{C})$ with an element in $\Lambda^{p+q}(T^* M \otimes \mathbb{C})$. They just look like different objects to me.