Relative Adjoint Algebras

Given a finite tensor category $\ca$, an exact indecomposable $\ca$-module category $\Mo$, and a tensor subcategory $\Do\subseteq \ca^*_\Mo$, we describe a way to produce \textit{exact} commutative algebras in the center $Z(\ca)$, measuring this inclusion. The construction of such algebras is done in an analogous way as presented by Shimizu \cite{Sh2}, but using instead the \textit{relative (co)end}, a categorical tool developed in \cite{BM} in the realm of representations of tensor categories. We provide some explicit computations.