  The subject of this paper is the higher structure of the strictification
adjunction, which relates the two fundamental bases of three-dimensional
category theory: the $\mathbf{Gray}$-category of $2$-categories and the
tricategory of bicategories. We show that -- far from requiring the full
weakness provided by the definitions of tricategory theory -- this adjunction
can be $\textit{strictly}$ enriched over the symmetric closed multicategory of
bicategories defined by Verity. Moreover, we show that this adjunction
underlies an adjunction of bicategory-enriched symmetric multicategories. An
appendix introduces the symmetric closed multicategory of pseudo double
categories, into which Verity's symmetric multicategory of bicategories embeds
fully.
