  We develop a 2-dimensional version of accessibility and presentability
compatible with the formalism of flat pseudofunctors. First we give
prerequisites on the different notions of 2-dimensional colimits, filteredness
and cofinality; in particular we show that sigma-filteredness and
bifilteredness are actually equivalent in practice for our purposes. Then, we
define bi-accessible and bipresentable 2-categories in terms of bicompact
objects and bifiltered bicolimits. We then characterize them as categories of
flat pseudofunctors. We also prove a bi-accessible right bi-adjoint functor
theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small bilex
2-categories and finitely bipresentable 2-categories. Finally, we show that
2-categories of pseudo-algebras of bifinitary pseudomonads on Cat are finitely
bipresentable, which in particular captures the case of Lex, the 2-category of
small lex categories. Invoking the technology of lex-colimits, we prove further
that several 2-categories arising in categorical logic (Reg, Ex, Coh, Ext, Adh,
Pretop) are also finitely bipresentable.
