let $f:X\to Y$ be a flat map between complex algebraic varieties and let $\mathcal F$ be a locally free sheaf on $X$. Very often I've seen the following map:
$$\wedge: \bigotimes^n (R^1f_{\ast}\mathcal F)\to R^nf_\ast\left(\wedge^n \mathcal F\right)$$
How is the map $\wedge$ formally defined? I don't understand it.
It seems that it is, somehow, canonically induced by the ordinary wedge product of locally free sheaves. But I don't understand why it ``commutes'' with the right derived functor $R^nf_{\ast}$.
Edit: I just wanted to add that my question is by coincidence related to the following unanswered question: link
Many thanks in advance