Left Kan extensions preserving finite products

We give conditions that ensure the preservation of finite products by left Kan extensions into cocomplete subcanonical sites. The conditions involve a suitable notion of flatness (interpreted in the internal logic of the site) of the functor that is extended and a good behavior of colimits used in the calculation of the left Kan extension. When the recipient category is a Grothendieck topos the good behavior of the colimits is granted and the flatness conditions turn out to be necessary as well as sufficient. We also show that the category of compactly generated Hausdorff spaces and that of small categories are equipped with suitable topologies that account for the well-known phenomenon of preservation of finite products by the geometric and categorical realization, respectively, of simplicial sets.