dual and tensor product of categories

Question

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)?