A morphism of sites is, unsurprisingly, the appropriate sort of morphism between sites. It is defined exactly so as to induce a geometric morphism between toposes of sheaves (or, more generally, exact completions). Let and be sites. A functor is a morphism of sites if is covering-flat, and preserves covering families, i.e. for every covering of an object , the family is a covering of . If has finite limits and all covering families in are strong epimorphisms, then covering-flatness of is...