Generalized equivariant model structures on $\mathbf{Cat}^I$

Let $I$ be a small category, $mathcal{C}$ be the category $mathbf{Cat}$, $mathbf{Ac}$ or $mathbf{Pos}$ of small categories, acyclic categories, or posets, respectively. Let $mathcal{O}$ be a locally small class of objects in $mathbf{Set}^I$ such that $mathrm{colim}_I O=*$ for every $Oin mathcal{O}$. We prove that $mathcal{C}^I$ admits the $mathcal{O}$-equivariant model structure in the sense of Farjoun, and that it is Quillen equivalent to the $mathcal{O}$-equivariant model structure on $mathbf{sSet}^I$. This generalizes previous results of Bohmann-Mazur-Osorno-Ozornova-Ponto-Yarnall and of May-Stephan-Zakharevich when $I=G$ is a discrete group and $mathcal{O}$ is the set of orbits of $G$.