Bill asks: Does every geometric morphism have right-derived functors on abelian objects? In principle, this would not require enough injectives since the universal property requested does not involve any specific kind of resolution. I think not. In my new Foreword to Abelian Categories on TAC (www.tac.mta.ca/tac/reprints/articles/3/foreword.pdf), specifically in the comments about pages 131-132, I recall the description of a locally small topos in which Ext(A,B) wants to have proper classes as values.