ScalingStacks

Definition 5.5 ([F1]). An $\mathcal{F}$-monoidal structure on an $\infty$-category $\mathcal{C}$ consists of a functor $p:\mathcal{F}\rightarrow{\rm Cat}_{\infty}$ with an identification $\mathcal{C}\simeq p(1_{*})$ such that the natural maps $p(J_{*})\rightarrow\prod_{J}p(1_{*})$ resulting from a choice of a point $x\in F(J)$ is an equivalence.