When $\mathcal{C}$ is symmetric monoidal closed, the Day convolution gives a symmetric monoidal closed structure to the category $[\mathcal{C},\mathbf{Set}]$.
Suppose that instead, it is $\mathcal{C}^{\text{op}}$ that is symmetric monoidal closed. Hence, $[\mathcal{C}^{\text{op}},\mathbf{Set}]$ is monoidal closed with the Day tensor product. My question is, can I use the Day convolution to say something about $[\mathcal{C},\mathbf{Set}]$? E.g., is it true that the opposite of it, $[\mathcal{C},\mathbf{Set}]^{\text{op}}$ is symmetric monoidal closed?
I guess this should be a matter of playing around with the definitions, but unfortunately I am not experienced enough to do this myself.
If $\mathcal{C}$ is monoidal then $\mathcal{C}^\mathrm{op}$ is monoidal. $[\mathcal{C}^{\mathrm{opop}},\mathsf{Set}] = [\mathcal{C},\mathsf{Set}]$ is monoidal closed under Day convolution, i.e. $[\mathcal{C},\mathsf{Set}]^\mathrm{op}$ is co-closed under Day convolution in the opposite category... if you want to be really explicit, then $$\begin{align} X \otimes Y(c) &= \int^{c_1,c_2 \in \mathcal{C}^\mathrm{op}} \mathcal{C}^\mathrm{op}(c,c_1 \otimes c_1) \times X(c_1) \times Y(c_2) \\ &= \int^{c_1,c_2 \in \mathcal{C}} \mathcal{C}(c_1\otimes c_2,c) \times X(c_1) \times Y(c_2) \end{align} $$
Did you have something else in mind?