Monoidal Adjunctions - Linearity and Duality

We explain two related constructions on the data of two monoidal symmetric closed categories $\mathscr{A}$ and $\mathscr{E}$ and monoidal functors $F: \mathscr{E}\to \mathscr{A}$ and $G: \mathscr{A}\to \mathscr{E}$. In a first part, we recall and partly extend work of A. Kock: In case $F$ is left-adjoint to $G$, and this adjunction is monoidal, we can equip the Eilenberg-Moore category $\mathscr{E}^T$ for $T$ being the canonical monad associated to the adjunction, with the structure of symmetric monoidal closed category, provided $\mathscr{E}$ has equalizers and $\mathscr{E}^T$ co-equalizers. In a second part, inspired by the Chu-construction, we build a category $\mathscr{R}_{G}$, which is symmetric monoidal closed as well, under the condition that $\mathscr{E}$ has pullbacks. Similarly we build a category $\mathscr{L}^{F}$ which is symmetric monoidal closed under the condition that $\mathscr{A}$ has what we call $F$-pushouts and $F$-pullbacks. In case $F \dashv G$ is a monoidal adjunction, we show that $\mathscr{L}^{F}$ and $\mathscr{R}_{G}$ are isomorphic as symmetric monoidal closed categories. We show also how $\mathscr{E}^T$ is related to both.