What is this map between tensor spaces called? (Change of coordinates).

Question

Let $V$ and $W$ be finite dimensional vector spaces. Let $A: V \to W$ be linear. Define the map $A^* : (W^*)^{\otimes r} \to (V^*)^{\otimes r}$ by $(A^*\alpha)(w_1, w_2, \ldots, w_r) = \alpha(Aw_1, Aw_2, \ldots Aw_r)$.

I have seen this called the adjoint but I do not think this is the common term. I also know that it has to deal with changes in coordinates but I am not sure why.

Answer 1

Pullback map, transpose map or dual map.