  In this paper, we propose an $\infty$-categorical definition of abstract
six-functor formalisms. Our definition is a variation on Mann's definition,
with the additional requirement of having Grothendieck and Wirthm\"uller
contexts, and recollements and descent for abstract blowups. Using Nagata's
compactification theorem, we show that such a six-functor formalism on
varieties can be given by just specifying adjoint triples on open immersions
and on proper maps, satisfying certain compatibilities. Descent for abstract
blowups and the existence of recollements are (almost) equivalent to a
hypersheaf condition for a Grothendieck topology on the category of ``varieties
and spans consisting of an open immersion and a proper map''. Using this
characterisation, we show that our category of six-functor formalisms is
equivalent to a subcategory of lax symmetric monoidal functors from the
category of smooth and complete varieties to the category of stable
$\infty$-categories and adjoint triples. Moreover, we can characterise which
lax symmetric monoidal functors on complete varieties, taking values in the
category of stable $\infty$-categories and adjoint triples, extend to
six-functor formalisms.
