On any ringed topos there is a notion of a quasi-coherent sheaf, see Definition Tag 03DL. The pullback of a quasi-coherent module via any morphism of ringed topoi is quasi-coherent, see Lemma Tag 03DO.

Let (X, O_X) be a scheme. Let tau = fppf, syntomic, etale, smooth, or Zariski. The site (Sch/X)_{tau} is a ringed site with sheaf of rings O. The category of quasi-coherent O_X-modules on X is equivalent to the category of quasi-coherent O-modules on(Sch/X)_{tau}, see Proposition Tag 03DX. This equivalence is compatible with pullback, but in general not with pushforward, see Proposition Tag 03LC.

Let me explain this last point a bit. SupposeĀ f : X —> Y is a quasi-compact and quasi-separated morphism of schemes. Denote f_{big} the morphism of big tau sites. Let F be a quasi-coherent O_X-module on X. The corresponding quasi-coherent O-module F^a on (Sch/X)_{tau} is given by the rule F^a(U) = Γ(U, φ^*F) if φ : U —> X is an object of (Sch/X)_{tau}. In general, for a sheaf G on (Sch/X)_{tau} we have f_{big, *}G(V) = G(V \times_Y X). Hence we see that the restriction of f_{big, *}F^a to V_{Zar} is given by the (usual) pushforward via the projection V \times_Y X —> V of the (usual) pullback of F to V \times_Y X via the other projection. It follows from the description of quasi-coherent sheaves on (Sch/Y)_{tau} as associated to usual quasi-coherent sheaves on Y that f_{big, *}F^a is quasi-coherent on (Sch/Y)_{tau} if and only if formation of f_*F commutes with arbitrary base change. This is simply not the case, even for morphisms of varieties, etc.

On the other hand, we know that f_*F commutes with any flat base change (still assuming f quasi-compact and quasi-separated). Hence f_{big, *}F^a is a sheaf H on (Sch/Y)_{tau} such that H|_{V_{Zar}} and H|_{V_{etale}} are quasi-coherent. Moreover, the same argument shows that if G is any sheaf of O-modules on (Sch/X)_{tau} such that G|_{U_{Zar}} or G|_{U_{etale}} is quasi-coherent for every U/X then H = f_{big, *}G is a sheaf such that H|_{V_{Zar}} or H|_{V_{etale}} are quasi-coherent for any object V of (Sch/Y)_{tau}. Moreover, this property is also preserved by f_{big}^* as this is just given by restriction.

Thus a convenient class of O-modules on (Sch/X)_{tau} appears to be the category of sheaves of O-modules F such that F|_{U_{etale}} is quasi-coherent for all U/X. These “quasi quasi-coherent sheaves” are preserved under any pullback and pushforward along quasi-compact and quasi-separated morphisms. Via the approach I sketched here they give a notion of quasi quasi-coherent sheaves on the tau site of any algebraic stack with arbitrary pullbacks and pushforward along quasi-compact and quasi-separated morphisms. An interesting example of a quasi quasi-coherent sheaf is the sheaf of differentials Ω on the etale site that I mentioned in here.

Can anybody suggest a better name for these sheaves?