I am a beginner in category theory, so the following question could be either obvious or of no interest. Anyhow, I cannot find a statisfactory framing, and possibly an answer for the following question:
Can we generalize the notion of 'linear' duality to categories other than $Vect$ or $A$-mod? Is this construction ever related to forming the opposite of a category (so to duality in the proper categorical sense)?
Obviously the problem is ill posed in this form, so I tried to work it out with my limited knowledge.
First of all, fix a category $\mathcal{C}$, then it always make sense to form the opposite category $\mathcal{C}^{op}$.
In some categories anyhow, there is a particular 'dualizing' object $D$, so that $Hom(\cdot,D)$ is a contravariant endofunctor. For instance in the category of $A$-modules (and in particular in $Vect$), the base ring (field) $A$ is such.
Q1 Is there a way to charachterize abstractly those categories and one object (or possibly more) therein, so that we have a way to perform this construction and it makes some sense? Are there at least other similarly relevant examples?
Obviously, a first obstruction to this is the fact that in general the $Hom$ functor is valued in $Set$ an not in our source category. But then there is also the 'meaningfulness' of this construction
Q2 In this situation, it may happen that the $Hom$ functor carries object to isomorphic ones, as in $FinVect$ though not naturally, and hence being contravariant reverses arrows but 'fixes' objects. Is there any relation to the formation of $\mathcal{C}^{op}$ in this case?
Thanks for any help or hints and references