Let $k$ be a field, and let $\mathcal{C}$ be a locally finitely presentable $k$-linear categories. The Deligne-Kelly tensor product $\mathcal{C}\boxtimes\mathcal{C}$ is the 2-universal locally finitely presentable $k$-linear category through which every bilinear functor out of $\mathcal{C}\times\mathcal{C}$ that is right exact in both variables factors.
If $\mathcal{C}$ has a monoidal structure $\otimes$ right exact in both variables, one can therefore consider the canonical right exact functor $T:\mathcal{C}\boxtimes \mathcal{C}\rightarrow \mathcal{C}$ induced by $\otimes$.
If $\otimes$ preserves compact objects and every compact object has both a left and a right dual in $\mathcal{C}$, is it true that $T$ has a left adjoint? (In fact, it is enough to prove that $T$ is left exact.)
