Ross Street, “Two-dimensional sheaf theory”
Ross Street, “Characterizations of bicategories of stacks”
These two papers basically supply our definitions of regular 2-category, coherent 2-category, exact 2-category, 2-pretopos, 2-site, 2-sheaf?, and Grothendieck 2-topos, as well as the proof of the 2-Giraud theorem. The definitions we are interested in are the bicategorical ones, which appear in the second paper, but the only proofs are in the first paper which does a strict version for strict 2-categories. Note in particular that where he uses the prefix “bi-” we use either “2-” or no prefix; thus our 2-sites are what he calles bisites, etc. There are also a couple of mistakes in the definitions and proofs which I have taken the liberty of correcting here.