  Univalent categories constitute a well-behaved and useful notion of category
in univalent foundations. The notion of univalence has subsequently been
generalized to bicategories and other structures in (higher) category theory.
Here, we zoom in on monoidal categories and study them in a univalent setting.
Specifically, we show that the bicategory of univalent monoidal categories is
univalent. Furthermore, we construct a Rezk completion for monoidal categories:
we show how any monoidal category is weakly equivalent to a univalent monoidal
category, universally. We have fully formalized these results in UniMath, a
library of univalent mathematics in the Coq proof assistant.
