Is there some appropriate setting where there is a notion of a "dual category," $\mathcal{C}^*$, and "tensor product of categories," $\mathcal{C}_1 \otimes \mathcal{C}_2$, such that we can identify the category of functors from $\mathcal{C}_1$ to $\mathcal{C}_2$ with the category $\mathcal{C}_1^* \otimes \mathcal{C}_2$ (analogous to the statement for vector spaces)?
dual and tensor product of categories
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category-theory
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0Well.. Note that $V^*\otimes W\cong\hom(V,W)$ is not widely general, even among the vector spaces, it only holds for *finite dimensional* $V$. The tensor product arises on actions of rings/monoids/categories. Vector spaces are *modules* over (i.e. actions on) fields/division rings. So, the analogy goes rather like rings~categories, (bi-)modules~profunctors, with tensor and hom. – 2017-01-18