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    $\begingroup$ There is a $C_2$-action on the category of finite-dimensional vector spaces and isomorphisms which sends a vector space to its dual and a linear isomorphism to the inverse of its transpose (this is a baby case of the cobordism hypothesis). A vector space with a symmetric non-degenerate bilinear form is a homotopy fixed point of this action, and antisymmetric forms have a similar description. In general, given an object of your category, a lift to homotopy fixed points need not exist, and if it does, need not be unique; that seems to encompass most of your examples. $\endgroup$ Commented Apr 22, 2021 at 21:45
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    $\begingroup$ The involution needed to decompose a bilinear form $\langle\cdot, \cdot\rangle$ into its symmetric and antisymmetric $\langle\cdot, \cdot\rangle_\pm$ parts is the "swap the arguments" involution. (I think this might be related to what @BertramArnold is saying, but I am not enough of a geometer really to be sure.) Namely, $\langle v, w\rangle_\epsilon = \frac1 2(\langle v, w\rangle + \epsilon\langle w, v\rangle)$. $\endgroup$ Commented Apr 22, 2021 at 22:20
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    $\begingroup$ If the question is just about specific objects, then the isomorphism of a finite dimensional vector space with its dual is no different than the isomorphism between any two vector spaces of the same dimension. What's interesting there is that the dual is functorial but the isomorphism is not natural. $\endgroup$ Commented Apr 22, 2021 at 22:58
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    $\begingroup$ @BenjaminSteinberg I keep oscillating between understanding and misunderstanding your comment. Certainly "no natural isomorphism" is more precise and focussed than "no canonical isomorphism". But on the other hand the functor you mention is contravariant, and there can be no natural transformations between identity and this functor. Or does it still make sense to talk about isomorphisms between covariant and contravariant self-equivalences?? $\endgroup$ Commented Apr 23, 2021 at 3:59
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    $\begingroup$ Related: Example of an unnatural isomorphism $\endgroup$ Commented Apr 25, 2021 at 18:43