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I don't think that there is a "canonical" choice, and I will point to another headache. Is there a " natural isomorphism $\det X\cong \det X^*$? If so, how does this work in the following situation.

Given a short exact sequence of finite vector spaces

$$ 0 \to A\to B\to C\to 0 $$

we get a dual short exact sequence

$$ 0\to C^* \to B^* \to A^*\to A^* \to 0. $$

For more details an a possible way out of this thorny situation see Section 1.2 of my book on torsion. There I explain in some detail Deligne's rules of operating with determinant lines.
show/hide this revision's text 1

I don't think that there is a "canonical" choice, and I will point to another headache. Is there a " natural isomorphism $\det X\cong \det X^*$? If so, how does this work in the following situation.

Given a short exact sequence of finite vector spaces

$$ 0 \to A\to B\to C\to 0 $$

we get a dual short exact sequence

$$ 0\to C^\to B^\to A^*\to 0. $$

For more details an a possible way out of this thorny situation see Section 1.2 of my book on torsion. There I explain in some detail Deligne's rules of operating with determinant lines.