A relationship between multiple conjugation quandle/biquandle colorings

We define a functor $\mathcal{Q}$ from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle $X$, there is a one-to-one correspondence between the set of $X$-colorings and that of $\mathcal{Q}(X)$-colorings diagrammatically for any handlebody-link and spatial trivalent graph. In particular, we prove that the set of $G$-family of Alexander biquandles colorings is isomorphic to that of $G$-family of Alexander quandles colorings as modules.