(SG) Subobjects of 1 Generate, and

(IC) locally, epimorphisms split (“Internal” axiom of Choice).

In particular, epimorphisms need not actually split, i.e., (AC) may fail, i.e., the implication (AC) => (IC) & (SG) is not reversible.

[NB: in contrast, for Grothendieck topoi, each of the implications (AC) => (IC) & (SG) => (B) & (SG)

The topos: locally finite sheaves over Heath’s

(Local finiteness => (IC); 0-dimensionality of the base => (SG).)

Same topos can then be used as counterexample to the reversibility of another couple of handfuls of implications among other conjunctions of similar topos-theoretic conditions.

[For details, consult that 30-year-old reference mentioned at the outset.]

[ Ref. ]