For an oriented knot $K$, we construct a functor from the category of pointed quandles to the category of quandles in three different ways. This functor-valued invariant of a knot is an extension of the knot quandle. We also extend the quandle cocycle invariants of knots by using these quandle-valued invariants, and study their properties.
Tetsuya ITO. "A functor-valued extension of knot quandles." J. Math. Soc. Japan 64 (4) 1147 - 1168, October, 2012. https://doi.org/10.2969/jmsj/06441147