Let $A$ be a $k$-algebra for a commutative ring $k$. Denote by $A^e$ the $k$-algebra $A\otimes A^{op}$. The center of a bimodule yields a functor $C$ from the category of right $A^e$-modules $Mod_{A^e}$ to the category of $k$-modules $Mod_k$. There are forgetful functors $U_1:{}_{A^e}Mod_{A^e\otimes A^e}\rightarrow Mod_{A^e}$ and $U_2:{}_{A^e}Mod_{A^e}\rightarrow Mod_k$ which each forget the left and right-most $A^e$-actions.
It is clear that the functor $C$ extends to a unique functor $\overline{C}:{}_{A^e}Mod_{A^e\otimes A^e}\rightarrow {}_{A^e}Mod_{A^e}$ such that $C\circ U_1=U_2\circ \overline{C}.$ Can this fact be explained more categorically, by a universal property or something like that?