  We prove a general theorem which includes most notions of "exact completion".
The theorem is that "k-ary exact categories" are a reflective sub-2-category of
"k-ary sites", for any regular cardinal k. A k-ary exact category is an exact
category with disjoint and universal k-small coproducts, and a k-ary site is a
site whose covering sieves are generated by k-small families and which
satisfies a weak size condition. For different values of k, this includes the
exact completions of a regular category or a category with (weak) finite
limits; the pretopos completion of a coherent category; and the category of
sheaves on a small site. For a large site with k the size of the universe, it
gives a well-behaved "category of small sheaves". Along the way, we define a
slightly generalized notion of "morphism of sites", and show that k-ary sites
are equivalent to a type of "enhanced allegory".

Category Theory