ScalingStacks

Proof. We apply Proposition 2.5 to $\mathcal{C}_{\infty}$ and $\mathcal{D}_{\infty}=\operatorname{N}\!\operatorname{Ch}\!^{b}(\mathcal{C}^{w=0}),$ the stable $\infty$-category of bounded chain complexes with values in $\mathcal{C}^{w=0}$ from [Lur09, Example 4.4.5.1] and [Lur12, Section 1.3.1]. The homotopy category $\operatorname{h}\!\mathcal{D}_{\infty}=\operatorname{K}^{b}(\mathcal{C}^{w=0})$ of $\mathcal{D}_{\infty}$ is the bounded homotopy category of chain complexes in $\mathcal{C}^{w=0}$ and equipped with the canonical weight structure with heart $\mathcal{C}^{w=0},$ see Example  2.2(1). The heart of the $\infty$-category $\mathcal{D}_{\infty}$ is $\mathcal{D}_{\infty}^{w=0}=\operatorname{N}\!(\mathcal{C}^{w=0}),$ the nerve of $\mathcal{C}^{w=0}.$

The unit of the adjunction between the nerve and homotopy category functors yields a functor $\epsilon:\mathcal{C}_{\infty}^{w=0}\to\mathcal{D}_{\infty}^{w=0}=\operatorname{N}\!(\operatorname{h}\!\mathcal{C}_{\infty}^{w=0}).$ The $\infty$-categorical weight complex functor $t_{\infty}:\mathcal{C}_{\infty}\to\mathcal{D}_{\infty}=\operatorname{N}\!\operatorname{Ch}\!^{b}(\mathcal{C}^{w=0})$ is the unique (up to equivalence) weight exact functor such that $\operatorname{Res}(t_{\infty})=\epsilon,$ where $\operatorname{Res}$ is the functor defined in Proposition   2.5. On the homotopy categories, $t_{\infty}$ induces the weight complex functor $t$. ∎