Is there a characterization of coverings in subcanonical pretopologies? Let $\mathcal C$ be a category.  A sieve for $\mathcal C$ is called strictly universally epimorphic if it is one of the covering sieves for the canonical topology on $\mathcal C$.  
SGA4 gives the following characterization of strictly universally epimorphic sieves (Proposition II-2.6):

A sieve $R\hookrightarrow h_X$ is strictly universally epimorphic if and only if for all morphisms $Y\to X$ in the sieve and for all objects $z$ of $\mathcal C$, the map
$$
\text{Hom}(Y,Z)\to\varprojlim_{\mathcal C/(Y\times_X R)} \text{Hom}(-,Z)
$$
is a bijection.

Is there a similar characterization of the covering families in pretopologies generating the canonical topology?  I noticed that many pretopologies have covering families of the form

Jointly surjective families of [...] morphisms

so I was hoping that something like 'jointly epimorphic families' would work, but this only seems to be enough to show that representable presheaves are separated.  
Vistoli's descent notes infer that

every fpqc morphism of schemes is an effective
  epimorphism in (Sch/S)

is equivalent to

the fpqc topology is subcanonical

which suggests that jointly effectively epimorphic (?) families might work.
 A: What Vistoli says is not entirely accurate. One needs a bit more to go from knowing what universally effective epimorphisms are to knowing what the sieves in the canonical topology are. Certainly what is true is that a morphism is a universally effective epimorphism if and only if it is a covering with respect to the canonical topology (at least assuming the existence of kernel pairs – but that's just because effective epimorphisms are usually only defined when kernel pairs exist).
Many topologies on categories of spaces can be constructed by choosing a class of "covering" morphisms. To make this precise requires a definition: an infinitary extensive category is a category in which coproducts of small families of objects exist and are disjoint and universal (in the same sense as in Giraud's axioms for a topos). The key point about infinitary extensive categories is this: for any small family $(X_i : i \in I)$, the coproduct injections comprise a covering family $\{ X_j \to \coprod_{i \in I} X_i : j \in I \}$ with respect to the canonical topology. For example, $\mathbf{Top}$ and $\mathbf{Sch}$ are infinitary extensive categories.
Now, given an infinitary extensive category, a superextensive topology is a Grothendieck topology such that coproduct injections give covering families as above. In general, one may speak of the smallest superextensive topology containing a class of "covering" morphisms, and we get these examples:


*

*The standard topology on $\mathbf{Top}$ is the smallest superextensive topology such that surjective local homeomorphisms are coverings.

*The Zariski topology (resp. étale, fppf, fpqc) on $\mathbf{Sch}$ is the smallest superextensive topology such that surjective local isomorphisms (resp. étale, fppf, fpqc morphisms) are coverings.


The fact that these topologies are subcanonical follows from the fact that the "covering" morphisms are universally effective epimorphisms. In particular, on any infinitary extensive category, there is a subcanonical superextensive topology generated by all universally effective epimorphisms, but it is not clear to me whether this is necessarily the same as the canonical topology. (This boils down to set theory: can every universally effective-epimorphic sieve be refined by one that is generated by a small family of morphisms? The answer is obviously yes if the category is small, but the only small infinitary extensive category is $\mathbb{1}$.)
