  We construct a cubical analogue of the rigidification functor from
quasi-categories to simplicial categories present in the work of Joyal and
Lurie. We define a functor from the category of cubical sets of
Doherty-Kapulkin-Lindsey-Sattler to the category of (small) simplicial
categories. We show that this rigidification functor establishes a Quillen
equivalence between the Joyal model structure on cubical sets (as it is called
by the four authors) and Bergner's model structure on simplicial categories. We
follow the approach to rigidification of Dugger and Spivak, adapting their
framework of necklaces to the cubical setting.
