Let $X$ and $Y$ be Artin algebraic stacks, and $f:X\to Y$ a morphism. I am interested in a pullback morphism
$Lf^\ast : D^-_{qcoh}(Y)\to D^-_{qcoh}(X)$.
In his paper Sheaves on Artin stacks, Martin Olsson uses the lisse-etale site to define $D_{qcoh}(X)$ and $D_{qcoh}(Y)$, and remarks that because $f:X\to Y$ does not induce a morphism of lisse-etale topoi, he can define $Lf^\ast$ only on a category of projective systems of derived category objects.
Question: Is this the best one can expect in the world of Artin stacks, or can we do better, for example, by replacing the lisse-etale site with the big etale site. I would hope so, because $f$ does induce a morphism of big etale topoi.
