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 $O\in \mathcal{O}$. 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 $G$.