My question is in reference to the derivation below from the physics site. It shows how the metric tensor raises and lowers indices. I cut it off halfway through because I didn't think the full derivation was relevant to the question, but here it is, just in case. I don't believe my question is a duplicate.
My question is, why is the map $\tau$ considered to be "natural"? Does the fact that the vector space has an inner product necessitate that the dual has one? (maybe so that there's an isomorphism?)
Edit: I'm asking if the fact that the vector space has an inner product induces the need for its dual to have an inner product to create an isomorphism.