Suppose $G$ is a finite algebraic group over a field $k$ with a split $BN$-pair, and Coxeter system $(W,S)$. The parabolic induction and restriction functors $R_{L}^G=kGe_U\otimes_{kL}-$ and $^\ast R^G_{L}=e_UkG\otimes_{kG}-$ are well known to be biadjoint. (Here I'm taking $L$ to be a Levi subgroup, and $U$ the unipotent of some parabolic subgroup $P$ containing $L$, and $e_U$ to be the idempotent $\frac{1}{|U|}\sum_{u\in U}u$ in $kG$.)
Viewing $R^G_L$ to be left adjoint to $^\ast R^G_L$ for instance, is there is explicit description of what the component morphisms of the unit and counit to the adjunction are?