ScalingStacks

Proposition 2.10. Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories with bounded weight structures and $F:\mathcal{C}\to\mathcal{D}$ be a weight exact functor. Assume that there is an enhancement to a functor $F:\mathcal{C}_{\infty}\to\mathcal{D}_{\infty}$ between stable $\infty$-categories. Then

1. (1)

   The following diagram of functors commutes up to natural isomorphism

2. (2)

   If $\mathcal{C}_{\infty}$ is symmetric monoidal and $\mathcal{C}^{w\geq 0}$and $\mathcal{C}^{w\leq 0}$ are closed with respect to the monoidal structures then the weight complex functor can be turned into a symmetric monoidal functor.