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](https://stackoverflow.com/questions/13366820/how-do-you-make-lettered-lists-using-markdown).)


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](https://nforum.ncatlab.org/discussion/3008/internally-projective-objects/) 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.



Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)
There is a counterexample, due to Todd Trimble, in [another nForum thread](https://nforum.ncatlab.org/discussion/4342/internally-projective-objects/?Focus=35649#Comment_35649); cf. the nLab entry on [internally projective objects](https://ncatlab.org/nlab/show/internally+projective+object). Theorem 2 of [loc. cit.](https://ncatlab.org/nlab/show/internally+projective+object) 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, 



> 
> [...] since if so, then every projective object would be internally projective, which as we show below is not the case.
> 
> 
> 


In fact, this is not "shown below", but a link is provided to [this counterexample](https://ncatlab.org/nlab/show/presentation+axiom#counter) (which is just the same as the one in the aforementioned [nForum entry by Todd Trimble](https://nforum.ncatlab.org/discussion/4342/internally-projective-objects/?Focus=35649#Comment_35649)). Note that by proposition 1 of the nLab entry, if the topos has enough projectives and projectives are closed under binary products, then every projective object is internally projective.


