Let $C, D$ be two (dg) categories and let $M$ be a $C-D$-bimodule, i.e. a right module over $C^{op}⊗ D$. I am trying to understand which directions $M$ induces functors and which of these functors are left/right adjoint.
Let $mod-C, C-mod$ be right, left $C$ modules respectively. Note that the Yoneda functor induces a functor $C→mod-C$ into right C modules; the contravariant Yoneda functor induces a functor $C^{op}\rightarrow mod-C^{op}= C-mod$ into left $C$ modules.
Then tensoring with the bimodule $M$ induces a functor
$$
\otimes_C M: mod-C → mod-D
$$
which has a right adjoint
$$
Hom_{mod-D}(M, \_): mod-D → mod-C
$$
The $C-D$ -bimodule $M$ also defines a $D^{op}-C^{op}$-bimodule $M^{op}$. Then tensoring with $M^{op}$ defines a functor $$ \otimes_{D^{op}} M^{op}: mod-D^{op}\rightarrow mod-C^{op} $$ with right adjoint $$ Hom_{mod-C^{op}}(M^{op},\_ ): mod-C^{op}\rightarrow mod-D^{op} $$ Taking $op$ of the last two functors, one gets $$ (\_\otimes_{D^{op}} M^{op})^{op}: (mod-D^{op})^{op} \rightarrow (mod-C^{op})^{op} $$ with left adjoint (since taking op switches left and right adjoints) $$ (Hom_{mod-C^{op}}(M^{op},\_ ))^{op}: (mod-C^{op})^{op}\rightarrow (mod-D^{op})^{op} $$ One reason for taking $op$ is that now there is a covariant Yoneda embedding $$ C\rightarrow (mod-C^{op})^{op} $$
So we have two functors $mod-C \rightarrow mod-D$ and $(mod-C^{op})^{op}\rightarrow (mod-C^{op})^{op}$, both of which have right adjoints.
Question: is there a natural way to get a functor $F$ on from a category of C-modules to a category of D-modules that has a left-adjoint?
Note that if $M$ is induced by an actual functor $F: C→ D$, then there is an extension $F: mod-C\rightarrow mod-D$ with right adjoint $F^*: mod-D\rightarrow mod-C$. Hence, $F^{op}: C^{op}→ D^{op}$ has an extension $F^{op}: mod-C^{op}\rightarrow mod-D^{op}$ with right adjoint $(F^{op})^*: mod-D^{op}\rightarrow mod-C^{op}$ and hence $(F^{op})^{op}: (mod-C^{op})^{op}\rightarrow (mod-D^{op})^{op})$ has left adjoint $((F^{op})^*)^{op}: (mod-D^{op})^{op}\rightarrow (mod-C^{op})^{op}$.
Question: how does one construct a similar left-adjoint in the case of a general $C-D$ bimodule $M$?