The free 2-category on a 1-category with respect to pseudo-functors into 2-categories

Question

Let $\mathbb{C}$ be an ordinary 1-category. I'm interested in the following potential construction. Is there a 2-category $\widetilde{\mathbb{C}}$ (equipped with a pseudo-functor $\eta : \mathbb{C} \to \widetilde{\mathbb{C}}$) with the universal property that, for any 2-category $\mathcal{K}$, pseudo-functors $\mathbb{C} \to \mathcal{K}$ correspond bijectively to 2-functors $\widetilde{\mathbb{C}} \to \mathcal{K}$? If so, is there an explicit description of this 2-category $\widetilde{\mathbb{C}}$, and/or is there a good reference in the literature for this construction?

Answer 1

I will write $\mathsf{BiCat}$ for the 1-category of weak 2-categories, and will write $\mathsf{StrBiCat}$ for the 1-category of strict 2-categories. There is a ''strictification'' functor $\mathrm{str}\colon \mathsf{BiCat}\to\mathrm{StrBiCat}$ that is left adjoint (in the 1-categorical sense) to the inclusion $\mathrm{StrBiCat}\hookrightarrow\mathrm{BiCat}$. That means that the construction you're looking for is $\widetilde{\mathbb{C}}=\mathrm{str}(\mathbb{C})$, with the pseudofunctor $\eta\colon\mathbb{C}\to\widetilde{\mathbb{C}}$ being given by the unit of the adjunction. Note that it is not necessary for $\mathbb{C}$ to be a 1-category: any weak 2-category would do. There is also an explicit (and rather simple) description of the strictification functor, for instance in $\S$4.10 in Gordon--Power--Street Coherence for Tricategories. They also point you towards earlier references.