Knot quandles vs. knot biquandles

As a generalization of quandles, biquandles have given many invariants of classical/surface/virtual links. In this paper, we show that the fundamental quandle [Formula: see text] of any classical/surface link [Formula: see text] detects the fundamental biquandle [Formula: see text]; more precisely, there exists a functor [Formula: see text] from the category of quandles to that of biquandles such that [Formula: see text]. Then, we can expect invariants from biquandles to be reduced to those from quandles. In fact, we introduce a right-adjoint functor [Formula: see text] of [Formula: see text], which implies that the coloring number of a biquandle [Formula: see text] is equal to that of the quandle [Formula: see text].