Lifting Enhanced Factorization Systems to Functor $2$-Categories

In this paper we give sufficient conditions for lifting an enhanced factorization system $(\mathcal{E},\mathcal{M})$ on a $2$-category $\mathbf{D}$ to the functor $2$-category ${\mathbf{D}}^{\mathbf{C}}$, where $\mathbf{C}$ is a small $2$-category. Due to previous work of Lack, our work provides coherence results for $2$-monads on functor $2$-categories. These coherence results are of immediate interest to an ongoing collaboration of Guillou, May, Merling, and Osorno on equivariant infinite loop space theory.