  Let $X$ be an algebraic stack with quasi-affine diagonal of finite type over
a field $k$ of characteristic $0$. We extend the well-known equivalence
$\mathsf{D}^+(\mathsf{QCoh}(X)) \simeq \mathsf{D}_{\mathrm{qc}}^+(X)$ to
unbounded derived categories. We also prove that if $X$ is smooth over $k$,
then $\mathsf{D}_{\mathrm{qc}}(X)$ is compactly generated. We accomplish the
former using the descendable algebras of Mathew. We also establish related
results in positive and mixed characteristics.
