ScalingStacks

Proof. The necessity of $M\otimes-$ preserving limits is noted above ($\mathcal{C}$ is closed by virtue of being presentable with monoidal structure distributing over colimits, see [L4, Proposition 2.1.12]). To demonstrate sufficiency, assume that $M\otimes-$ preserves limits, and then consider the endofunctor of $\mathcal{C}$ defined by tensoring with $M$. By assumption on $M$ and $\mathcal{C}$, this functor preserves all limits and colimits. We may now apply the adjoint functor theorem of [L2] to deduce the existence of a left adjoint $F$ to $M\otimes-$. Denote by $M^{\vee}$ the value $F(1_{\mathcal{C}})$ of $F$ applied to the unit of $\mathcal{C}$. The existence of unit and trace maps $1_{\mathcal{C}}\rightarrow M\otimes M^{\vee}\rightarrow 1_{\mathcal{C}}$ is now a particular instance of the unit and counit maps for this adjunction, which implies that $M$ and $M^{\vee}$ are in duality. Hence $M$ is dualizable.

∎