Remarks on exactness notions pertaining to pushouts
  We call a finitely complete category diexact if every Mal'cev relation admits a pushout which is stable under pullback and itself a pullback. We prove three results relating to diexact categories: firstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only if it is Barr-exact, and every pair of monomorphisms admits a pushout which is stable and a pullback; and thirdly, that a small category with finite limits and pushouts of Mal'cev spans is diexact if and only if it admits a full structure-preserving embedding into a Grothendieck topos. 
Richard Garner
math.CT