Left exact presheaves on a small pretopos (Q1295689)

Let \({\mathcal E}\) be a small category with finite colimits. A left exact presheaf on \({\mathcal E}\) is a contravariant functor from \({\mathcal E}\) to the category \({\mathcal S}et\). They form with natural transformations the category \({\mathcal L}ex({\mathcal E})\). In this paper it is proved that whenever \({\mathcal E}\) is exact then \({\mathcal L}ex({\mathcal E})\) is exact precisely when in \({\mathcal E}\) the equivalence relation generated by a reflexive symmetric relation \(R\) is a finite iterate of \(R\). When this condition is satisfied and moreover \({\mathcal E}\) is a pretopos, then \({\mathcal L}ex({\mathcal E})\) is a topos. Various examples and counterexamples are given.