Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)

Lemma D4.5.3 of Johnstone’s Sketches of an Elephant states:

Lemma. For an object $A$ of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent:

1. $A$ is internally projective [i.e. $\Pi_A : \E/A \to \E$ preserves epimorphisms];
2. $(−)^A : \E \to \E$ preserves epimorphisms;
3. for every epimorphism $\newcommand{\epito}{\twoheadrightarrow} e : B \epito A$ in $\E$, there exists $C \epito 1$ such that $C^∗(e)$ is split epic.

(These conditions should be (i), (ii), (iii), but lettered lists don’t seem to be available in this Markdown dialect.)


However, we couldn’t find a counterexample, and the nforum thread doesn’t give one either, so the question is still a bit unsettled. Does anyone know either a counterexample to the implication (iii) $\Imp$ (i), or else an argument that it holds?

To keep any discussion clear, I suggest the term weakly projective for condition (iii) (unless someone knows a more established term for it). As Mike says in the nforum thread, if you strengthen this to its stable version, by quantifying over epis $e : B \to U \times A$ for all objects $U$, then it does imply internal projectivity.

There is a counterexample, due to Todd Trimble, in another nForum thread; cf. the nLab entry on internally projective objects. Theorem 2 of loc. cit. tells that for an object $A$ to be internally projective, it is necessary and sufficient that for all objects $B$ and $U$ and every epimorphism $e\colon B\to A\times U$ there is an epimorphism $f\colon C\to U$ such that the pull back morphism $(\operatorname{id}_A\times f)^*(e)\colon (\operatorname{id}_A\times f)^*B\to A\times C$ is split. Right after the proof it is noted that in Lemma 4.5.3 of the Elephant, the third statement is just the the special case $U=1$ and that this is insufficient to prove that $A$ is internally projective,