Filtered colimits in $D(X_{\text{proét}})$

Question

In §5.2 of their pro-étale paper, Bhatt and Scholze define a functor $L$ in terms of a sequential colimit, and, separately, a full subcategory $D’$ of $D(X_{\text{proét}}):=D(X_{\text{proét}},\mathbf Z)$ as the smallest triangulated subcategory of $D(X_{\text{proét}})$ containing the image of $D(X_{\text{ét}})$ and closed under filtered colimits. I believe $L$ can actually be defined by the corresponding homotopy colimit, but implicit in both these definitions is that $D(X_{\text{proét}})$ has all filtered colimits, and I have no idea why that would be true; I’m not used to computing filtered colimits in triangulated categories. Whether they mean by $D(X_{\text{proét}})$ the triangulated category or the corresponding stable $(\infty,1)$-category, why would either have all filtered colimits?

Answer 1

I haven't looked at the paper in a long time, but they surely mean the $\infty$-category--the cokernel functor $M$ would not even be functorial on the plain derived 1-category, and certainly this category does not have filtered colimits. The derived $\infty$-category of any Grothendieck topos is locally presentable so definitely has all colimits.