Relative Adjoint Algebras

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