If a category $\mathcal{C}$ is (symmetric) monoidal closed, is the opposite category $\mathcal{C}^{\text{op}}$ also monoidal closed?
It is not clear to me whether by dualising the natural bijection $$\mathcal{C}(a\otimes b,c)\cong\mathcal{C}(a,[b,c])$$ we get a monoidal closed structure on $\mathcal{C}^{\text{op}}$.
From either definition of adjunction it follows that $F ⊣ G$ iff $G^{\mathrm{op}} ⊣ F^{\mathrm{op}}$. For a monoidal category to be monoidal closed it is enough that every $- ⊗ B$ be left adjoint. The dual of this is that $- ⊗ B$ is right adjoint. So no, the dual of a monoidal closed category is not itself closed (instead it is "coclosed", although this is not an often used term).
In fact, the dual of a Cartesian closed category is never closed, unless the category is the singleton: see here for a short proof.
