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:

- $A$ is internally projective [i.e. $\Pi_A : \E/A \to \E$ preserves epimorphisms];
- $(−)^A : \E \to \E$ preserves epimorphisms;
- 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.)

The implications (i) $\newcommand{\Iff}{\Leftrightarrow} \Iff$ (ii) and (ii) $\newcommand{\Imp}{\Rightarrow}\Imp$ (iii) are fine, but (iii) $\Imp$ (i) is rather murky. The argument given in the Elephant is a very brief sketch; I and a couple of colleagues spent some time today trying to figure out the details, couldn’t, and ended up doubting that this implication is correct. Searching around, this nforum thread shows that Mike Shulman and Jonas Frey have previously come to essentially the same conclusion.

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.