Let consider a closed symmetric monoidal cateogry, $\mathscr C,\otimes$, with adjunction $(X\otimes-)\dashv\mathrm{Hom}(X,-)$ for all objects $X$.

The following isomorphis, valid in categories such as sets of vector spaces, over a field, $$\mathrm{Hom}(\coprod_iX_i,Y)\cong\prod_i\mathrm{Hom}(X_i,Y)$$ leads to the following question:

The contravariant internal Hom functor $\mathrm{Hom}(-,Y)$ is also adjoint?


For a symmetric (or even just braided) monoidal category, yes.

Proof: $$\begin{align} \mathcal{C}(A,\text{Hom}(B,C)) & \cong \mathcal{C}(A\otimes B,C) \\ & \cong \mathcal{C}(B \otimes A, C)\text{ via braiding } \\ & \cong \mathcal{C}(B, \text{Hom}(A,C)) \end{align}$$ So, $\text{Hom}(-,C)$ is adjoint to itself on the right, just as for cartesian closure.


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