Derived pullback of complexes with quasi-coherent cohomology sheaves exists in general.
Proposition 104.7.1. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. The exact functor f^* induces a commutative diagram
\xymatrix{ D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \ar[r] & D(\mathcal{O}_\mathcal {X}) \\ D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}) \ar[r] \ar[u]^{f^*} & D(\mathcal{O}_\mathcal {Y}) \ar[u]^{f^*} }
The composition
D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}) \xrightarrow {f^*} D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \xrightarrow {q_\mathcal {X}} D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})
is left derivable with respect to the localization D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}) \to D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y}) and we may define Lf^*_\mathit{QCoh} as its left derived functor
Lf_\mathit{QCoh}^* : D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})
(see Derived Categories, Definitions 13.14.2 and 13.14.9). If f is quasi-compact and quasi-separated, then Lf^*_\mathit{QCoh} and Rf_{\mathit{QCoh}, *} satisfy the following adjointness:
\mathop{\mathrm{Hom}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(Lf^*_\mathit{QCoh}A, B) = \mathop{\mathrm{Hom}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y})}(A, Rf_{\mathit{QCoh}, *}B)
for A \in D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y}) and B \in D^{+}_\mathit{QCoh}(\mathcal{O}_\mathcal {X}).
Proof.
To prove the first statement, we have to show that f^*E is an object of D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) for E in D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}). Since f^* = f^{-1} is exact this follows immediately from the fact that f^* maps \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) into \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) by Cohomology of Stacks, Proposition 103.8.1.
Set \mathcal{D} = D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}). Let S be the collection of morphisms in \mathcal{D} whose cone is an object of D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}). Set \mathcal{D}' = D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}). Set F = q_\mathcal {X} \circ f^* : \mathcal{D} \to \mathcal{D}'. Then \mathcal{D}, S, \mathcal{D}', F are as in Derived Categories, Situation 13.14.1 and Definition 13.14.2. Let us prove that LF(E) is defined for any object E of \mathcal{D}. Namely, consider the triangle
E' \to E \to P \to E'[1]
constructed in Lemma 104.5.4. Note that s : E' \to E is an element of S. We claim that E' computes LF. Namely, suppose that s' : E'' \to E is another element of S, i.e., fits into a triangle E'' \to E \to P' \to E''[1] with P' in D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}). By Lemma 104.5.4 (and its proof) we see that E' \to E factors through E'' \to E. Thus we see that E' \to E is cofinal in the system S/E. Hence it is clear that E' computes LF.
To see the final statement, write B = q_\mathcal {X}(H) and A = q_\mathcal {Y}(E). Choose E' \to E as above. We will use on the one hand that Rf_{\mathit{QCoh}, *}(B) = q_\mathcal {Y}(Rf_*H) and on the other that Lf^*_\mathit{QCoh}(A) = q_\mathcal {X}(f^*E').
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(Lf^*_\mathit{QCoh}A, B) & = \mathop{\mathrm{Hom}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(q_\mathcal {X}(f^*E'), q_\mathcal {X}(H)) \\ & = \mathop{\mathrm{colim}}\nolimits _{H \to H'} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {X})}(f^*E', H') \\ & = \mathop{\mathrm{colim}}\nolimits _{H \to H'} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {Y})}(E', Rf_*H') \\ & = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {Y})}(E', Rf_*H) \\ & = \mathop{\mathrm{Hom}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y})}(A, Rf_{\mathit{QCoh}, *}B) \end{align*}
Here the colimit is over morphisms s : H \to H' in D^+_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) whose cone P(s) is an object of D^+_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}). The first equality we've seen above. The second equality holds by construction of the Verdier quotient. The third equality holds by Cohomology on Sites, Lemma 21.19.1. Since Rf_*P(s) is an object of D^+_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}) by Proposition 104.6.1 we see that \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {Y})}(E', Rf_*P(s)) = 0. Thus the fourth equality holds. The final equality holds by construction of E'.
\square
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