The monoidal center construction and bimodules

Let image be a cocomplete monoidal category such that the tensor product in image preserves colimits in each argument. Let A be an algebra in image. We show (under some assumptions including “faithful flatness” of A) that the center of the monoidal category image of A–A-bimodules is equivalent to the center of image (hence in a sense trivial): image. Assuming A to be a commutative algebra in the center image, we compute the center image of the category of right A-modules (considered as a subcategory of image using the structure of image. We find image, the category of dyslectic right A-modules in the braided category image.