Suppose I have $n$ sequences, each of length $m$, and a function $f$ that takes one element from each sequence and returns a value. If I then form a sequence of like this
$$ (f(x^1_1,...,x^n_1),...,f(x^1_m,...,x^n_m)), $$
where $x^i_j$ is the $j^{th}$ element of the $i^{th}$ sequence, this seems like some kind of product to me. For example, similar to a dot product, with multiplication replaced by a $f$ and without summing up the resulting sequence.
Does this product have a good name?
I wouldn't call this a product. It's more of a lookup function.
All it appears to do is pull out one of the elements from one of the sequences.