  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.
