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There's a canonical way of going the other way, starting with two linear categories with nice finiteness properties, with adjoint functors between them and getting a pair of vector spaces with adjoint linear transformations. The vector spaces are generated by formal symbols for each object in the category, and the inner product between any objects is the dimension of the Hom space (so Hom spaces had better be finite dimensional). Note that this doesn't have to be symmetric.

Functors give linear transformations, and adjoint functors are adjoint in the usual sense.

You can soup up this construction when you have some more structures on your category. For example, if you have a direct sum, then you can impose the relation [A+B]=[A]+[B], $[A+B]=[A]+[B]$, and everything will work fine.

If your category is abelian, you can take Grothendieck group, where [A]+[C]=[B] $[A]+[C]=[B]$ for every short exact sequence $0\to A -> \to B -> C\to C\to 0$, but then you have to be much more careful about the fact that lots of functors (including Hom with objects in the category!) aren't exact: they don't send short exact sequences to short exact sequences. You need to use derived functors to fix this.

There's no canonical way of going the direction you asked, though in practice we have a very good record of being able to and I don't know of any really good examples of there being two equal natural seeming but different such constructions.

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There's a canonical way of going the other way, starting with two linear categories with nice finiteness properties, with adjoint functors between them and getting a pair of vector spaces with adjoint linear transformations. The vector spaces are generated by formal symbols for each object in the category, and the inner product between any objects is the dimension of the Hom space (so Hom spaces had better be finite dimensional). Note that this doesn't have to be symmetric.

Functors give linear transformations, and adjoint functors are adjoint in the usual sense.

You can soup up this construction when you have some more structures on your category. For example, if you have a direct sum, then you can impose the relation [A+B]=[A]+[B], and everything will work fine.

If your category is abelian, you can take Grothendieck group, where [A]+[C]=[B] for every short exact sequence A -> B -> C, but then you have to be much more careful about the fact that lots of functors (including Hom with objects in the category!) aren't exact: they don't send short exact sequences to short exact sequences. You need to use derived functors to fix this.

There's no canonical way of going the direction you asked, though in practice we have a very good record of being able to and I don't know of any really good examples of there being two equal natural seeming but different such constructions.