  We capture in the context of lex colimits, introduced by Garner and Lack, the
universal property of the free regular and Barr-exact completions of a weakly
lex category. This is done by introducing a notion of flatness for functors
$F\colon\mathcal C\to\mathcal E$ with lex codomain, and using this to describe
the universal property of free $\Phi$-exact completions in the absence of
finite limits, for any given class $\Phi$ of lex weights. In particular, we
shall give necessary and sufficient conditions for the existence of free
lextensive and free pretopos completions in the non-lex world, and prove that
the ultraproducts, in the categories of models of such completions, satisfy an
universal property.
