The monoidal structure of strictification

We study the monoidal structure of the standard strictification functor $\text{st}:\mathbf{B}\mathbf{i}\mathbf{c}\mathbf{a}\mathbf{t}\to \mathbf{2}\mathbf{C}\mathbf{a}\mathbf{t}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.