Center of a bimodule

Question

If we have a $\mathbb{k}$-algebra $R$ over a field $\mathbb{k}$ together with an $R$-bimodule $M$ we can consider the center $Z(M)=\{m\in M \vert \ \ rm=mr \ \ \forall r\in R\}.$ This is again an $R$-bimodule. If we consider $R$ as the regular bimodule, then $Z(R)\cong \operatorname{Hom}_{R-R}(R,R)$ as $\mathbb{k}$-algebras. Can we describe $Z(M)$ similarly? Is $Z(M)\cong \operatorname{Hom}_{R-R}(M,M)$ as $R$-bimodules? Can $Z(M)$ be described in more categorical terms?