ScalingStacks

Proposition 2.5. Let $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}$ be stable $\infty$-categories. Assume that their homotopy categories $\mathcal{C}=\operatorname{h}\!\mathcal{C}_{\infty}$ and $\mathcal{D}=\operatorname{h}\!\mathcal{D}_{\infty}$ are equipped with bounded weight structures. The hearts $\mathcal{C}_{\infty}^{w=0}$ and $\mathcal{D}_{\infty}^{w=0}$ are the full subcategories of $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}$ consisting of all objects in $\mathcal{C}^{w=0}$ and $\mathcal{D}^{w=0},$ respectively. Then restriction gives an equivalence of categories

$$\operatorname{Res}:\operatorname{Fun}^{\operatorname{w-ex}}(\mathcal{C}_{\infty},\mathcal{D}_{\infty})\to\operatorname{Fun}^{\operatorname{add}}(\mathcal{C}_{\infty}^{w=0},\mathcal{D}_{\infty}^{w=0})$$

between the $\infty$-categories of exact functors from $\mathcal{C}_{\infty}$ to $\mathcal{D}_{\infty}$ that induce weight exact functors from $\mathcal{C}$ to $\mathcal{D}$ and of additive functors between the hearts $\mathcal{C}_{\infty}^{w=0}$ and $\mathcal{D}_{\infty}^{w=0}.$