ScalingStacks

Definition 5.6. Let $\mathcal{C}$ be an $\mathcal{F}$-category. An $\mathcal{F}$-$\mathcal{C}$-module structure on an $\infty$-category $\mathcal{M}$ is a functor $q:\mathcal{F}_{+}\rightarrow{\rm Cat}_{\infty}$ extending the $\mathcal{F}$-monoidal structure on $\mathcal{C}$, together with an identification $q(+)\simeq\mathcal{M}$ such that the natural map $q((I\amalg+)_{*})\rightarrow\mathcal{C}^{I}\times\mathcal{M}$ is an equivalence for any $I$.