Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a biequivalence $\eta_{\mathcal{B}}: \mathcal{B} \rightarrow \textbf{st}\mathcal{B}$ such that, for any $2$-category $\mathcal{C}$, precomposition with $\eta_{\mathcal{B}}$ induces a bijection between $2$-functors from $\textbf{st}\mathcal{B}$ to $\mathcal{C}$ and pseudofunctors from $\mathcal{B}$ to $\mathcal{C}$.
Further, Corollary 3.6 in Alexander Campbell's article "How strict is strictification?" states that this bijection extends to an isomorphism $\textbf{Gray}(\textbf{st}\mathcal{B},\mathcal{C}) \simeq \textbf{Bicat}(\mathcal{B},\mathcal{C})$ of $2$-categories. The left-hand side is the $\operatorname{Hom}$–$2$-category in the $\textbf{Gray}$-category $\textbf{Gray}$, and thus consists of $2$-functors, pseudonatural transformations and modifications; the right-hand side is the $\operatorname{Hom}$–$2$-category in the tricategory of bicategories, pseudofunctors, pseudonatural transformations and modifications.
I am interested in the generalization of the above $2$-isomorphism to the setting of $\mathcal{V}$-bicategorical structures, where $\mathcal{V}$ is a closed, symmetric, (co)complete monoidal category.
More precisely, let $\mathcal{V}$–$\textbf{Cat}$ be the closed monoidal $2$-category of (small) $\mathcal{V}$-categories, $\mathcal{V}$-functors and $\mathcal{V}$-transformations. Following the definitions given by Garner and Shulman in "Enriched categories as a free cocompletion", we obtain the notions of bicategories, pseudofunctors, pseudonatural transformations and modifications enriched in $\mathcal{V}$–$\textbf{Cat}$ (as well as their strict variants), assembling into a tricategory $\mathcal{V}$–$\mathbf{Bicat}$.
We may now ask whether the above coherence results for bicategories also hold for $\mathcal{V}$-bicategories: given a $\mathcal{V}$-bicategory $\mathcal{B}$, is there a $\mathcal{V}$–$2$-category $\textbf{st}\mathcal{B}$ together with a $\mathcal{V}$-biequivalence $\eta_{\mathcal{B}}: \mathcal{B} \rightarrow \textbf{st}\mathcal{B}$ giving rise to $\mathcal{V}$–$2$-isomorphisms between $\mathcal{V}$–$\textbf{Gray}(\textbf{st}\mathcal{B},\mathcal{C})$ and $\mathcal{V}$–$\textbf{Bicat}(\mathcal{B,C})$, for any $\mathcal{V}$–$2$-category $\mathcal{C}$? If this is the case, is there a reference for the statement (preferably with some sketch of proof)? Or is it perhaps generally accepted as folklore?
I would also be interested in less general statements. For instance, if I understand correctly, in the bicategorical setting this statement is somewhat simpler if we assume $\mathcal{C} = \textbf{Cat}$. Is the above statement true in the case $\mathcal{C} = \mathcal{V}$–$\textbf{Cat}$? (The $\mathcal{V}$-pseudofunctors we consider then become a special case of modules studied in "Enriched categories as a free cocompletion")
Restricting further, is either of the statements (general $\mathcal{C}$ or $\mathcal{C} = \mathcal{V}$–$\textbf{Cat}$) true if $\mathcal{B}$ is a $\mathcal{V}$-monoidal category, perhaps by some general results on pseudomonoids in monoidal $2$-categories? Or even further, is any of the above statements true in the case when $\mathcal{V}$ is locally presentable?
Best Answer
In section 4 of my paper Not every pseudoalgebra is equivalent to a strict one, I sketched a proof that for any monoidal 2-category $\mathcal{W}$ with small sums preserved on both sides by its tensor product (which includes $\mathcal{V}\text{-Cat}$ for nice enough $\mathcal{V}$), there is a 2-monad on the 2-category of $\mathcal{W}$-graphs whose strict algebras and pseudo algebras are, respectively, strict $\mathcal{W}$-categories and (unbiased) $\mathcal{W}$-bicategories.
Moreover, under appropriate hypotheses on $\mathcal{W}$ (one of which, the existence of an appropriate factorization system, holds for $\mathcal{V}\text{-Cat}$), the general 2-monadic coherence theorem holds for this 2-monad. This means that the inclusion of the 2-category of pseudoalgebras into that of strict algebras has a strict left 2-adjoint, and the components of the adjunction unit are equivalences (thus every $\mathcal{W}$-bicategory is equivalent to its strictification). This universal property lies in between the two mentioned in your first two paragraphs: instead of an isomorphism of 2-categories of strict/pseudo functors, pseudonatural transformations, and modifications, we have an isomorphism of 1-categories of strict/pseudo functors and icons, which in particular entails a bijection between sets of strict/pseudo functors.
One could probably start from this and then work to extend it to pseudonatural transformations and modifications. I don't know offhand a reference that does this.