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I have split morphism of sites off from site, and in the process made bold to change the definition to the one I think is inarguably correct, using covering-flatness rather than representable-flatness (which is equivalent in case the sites have finite limits and subcanonical topologies). I made corresponding changes to the statements of the theorems relating morphisms of sites to geometric morphisms, but I don’t have the time or energy to change the proofs, so I left them as-is with pointers to my paper for proofs in the general case. It would be nice to have proofs of these theorems in the general case that don’t require the reader to understand $\kappa$-ary exact completion for general $\kappa$, but I can’t think when I might have time to write out such proofs in the foreseeable future myself.
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