TITLE: Are equivalences of categories stable under filtered colimits?
QUESTION [13 upvotes]: Let $D \colon \mathbf{J} \to \mathbf{Cat}$ be a filtered diagram of categories and functors. It has a colimit $\mathbf{C} = \mathrm{colim}\;D$. If you replace the diagram by a naturally isomorphic one $D' \colon \mathbf{J} \to \mathbf{Cat}$, then the colimit
$\mathbf{C'} = \mathrm{colim}\;D'$ is isomorphic to $\mathbf{C}$. What happens if you replace the diagram by a naturally equivalent one $D'' \colon \mathbf{J} \to \mathbf{Cat}$, is the colimit $\mathbf{C''} = \mathrm{colim}\;D''$ still equivalent to $\mathbf{C}$?
(I'm interested in $\mathbf{Cat}$, or in fact the category $\mathbf{MonCat}$ of monoidal categories, but suspect that there is a general statement in weak 2-categories. Any references are appreciated.)

REPLY [6 votes]: With regards the question about MonCat, I will assume that you are taking strong monoidal functors as morphisms.  (If you are dealing with strict morphisms things are straightforward.)
Then MonCat is an accessible category with filtered colimits and U:MonCat--->Cat preserves them.  Since, viewing it as a 2-functor, it also reflects equivalences this reduces the MonCat question to the Cat one.
More generally, if T is a finitary flexible (aka cofibrant) 2-monad on Cat, the 2-category of algebras and pseudomaps T-Alg is accessible with filtered colimits and U:T-Alg-->Cat preserves them.  This is a special case of Corollary 7.3 of my paper  https://arxiv.org/abs/2003.06375.  Flexibility or cofibrancy of the 2-monad means that it describes categorical structures with no equations between objects (like monoidal categories but not strict monoidal categories).  For strict structures like strict monoidal categories, it is not true.
As to why filtered colimits of equivalences are equivalences in Cat, the approach via combinatorial model categories described by Simon Henry is a good general approach.  An argument avoiding model categories, though similar in spirit, is to observe that, viewed as objects of the arrow category of Cat, equivalences are finitary injectives - see Diagram 2.3 of https://arxiv.org/abs/1712.02523 - and then use that finitary injectivity classes are always closed under filtered colimits, which is easy to prove (this is an easy part of the proof of Prop 4.7 of Locally presentable and accessible categories.)
P.S. Perhaps the original result of this nature is Lemma 5.4.9 of "Accessible categories .." by Makkai and Pare which shows that filtered colimits are bicolimits in Cat.  This implies the property about equivalences.