When an antinormal pair with identity composite fails to give a di-extension

Decide whether there exist pointed categories in which an antinormal pair (e,u) with the composite e u equal to the identity morphism need not yield a di-extension.

Background

Di-extensions are 3×3-diagrams whose rows and columns are short exact. The construction relates to ‘dinversion’ between antinormal pairs. The exercise probes the limits of this construction in pointed categories when the antinormal composite is the identity.