Let $\mathcal{C}$ be a locally presentable (∞,1)-category. A functor
(from the opposite of the homotopy category of $\mathcal{C}$ to Set)
is called a Brown functor if
it sends small coproducts to products;
it sends (∞,1)-pushouts in $\mathcal{C}\to Ho(\mathcal{C})$ to weak pullbacks in Set.
A weak pullback is a diagram that satisfies the existence clause of a pullback, but not necessarily the uniqueness condition. Hence the second clause in def. 1 says that for a (∞,1)-pushout square
in $\mathcal{C}$, then the induced universal morphism
into the actual pullback is an epimorphism.
(Brown representability theorem)
Let $\mathcal{C}$ be a locally presentable (∞,1)-category.
If
$\mathcal{C}$ is generated by a set
of compact objects (i.e. every object of $\mathcal{C}$ is an (∞,1)-colimit of the objects $S_i$.)
each $S_i$ admits the structure of a cogroup object in the homotopy category $Ho(\mathcal{C})$,
then a functor
(from the opposite of the homotopy category of $\mathcal{C}$ to Set)
is representable precisely if it is a Brown functor, def. 1.
A right Brown functor between categories of fibrant objects is a functor that preserves weak equivalences between fibrant objects.
Dually for a left Brown functor (for instance between Waldhausen categories).