Totally distributive toposes

A locally small category E is totally distributive (as defined by Rosebrugh and Wood) if there exists a string of adjoint functors , where is the Yoneda embedding. Saying that E is lex totally distributive if, moreover, the left adjoint t preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the essential subtoposes of presheaf toposes, studied by Kelly and Lawvere and by Kennett, Riehl, Roy, and Zaks.