maybe this is a stupid question, but I could not solve it after some time of meditation.
There are four different notions of regular categories:
1) A cartesian category with coequalizers of kernel pairs, where regulars epimorphisms (coequalizers of some parallel arrows) are stable under pullbacks;
2) A cartesian category with images, where extremal epimorphisms (covers http://ncatlab.org/nlab/show/extremal+epimorphism) are stable under pullback
3) A cartesian category with coequalizers, where regular epimorphisms are stable under pullbacks.
4) A category with images, where $g^{*} (\text{Im}f) \cong \text{Im}(g^{*}f)$ holds.
In Johnstone's elephant book ,it's asserted that 2 is equivalent to 4 and it's proved that 3 is equivalent to 2 + "existence of coequalizers". But in http://ncatlab.org/nlab/show/regular+category it's asserted that 1 is equivalent to 4 + "cartesianness" and it seems very strange. I doubt this is really true if Johnstone's assertion is true. So the question are:
Is 1 equivalent to 2?
Is 2 equivalent to 4?
Is 1 equivalent to 4 + "cartesianness"?
Is 1 equivalent to 3?
Thanks in advance.