I need to use the following result about essential points from Johnstone's Topos Theory:
Let $\mathcal{E}$ be a Grothendieck toposes and $\mathcal{S}$ be a topos with a natural number object. Show that a functor $\phi:\mathcal{E} \to \mathcal{S}$ has a left adjoint iff it is representable. Deduce that $\phi$ is the inverse image of an essential point of $\mathcal{E}$ iff it has the form $\mathrm{Hom}_{\mathcal{E}}(P,-)$ where $P$ is projective, connected and not isomorphic to $0$.
I personally am only interested in the case that $\mathcal{S}$ is Set for the time being. The representability part is standard, and being non-zero/connected correspond respectively to preserving $0$/coproducts. At first I assumed the aim was to show that projectivity of $P$ corresponds to preservation of coequalizers, and then use the special adjoint functor theorem to show that an adjoint exists.
However, being projective only guarantees preservation of epimorphisms. Of course, all epis in $\mathcal{E}$ and $\mathcal{S}$ are regular, so being projective is necessary to preserve coequalizers; also kernel pairs are preserved, so coequalizers of kernel pairs are preserved. However, after playing around for some time (see below) I cannot see why coequalizers of arbitrary pairs should be preserved.
The confusing thing is that there is a hint in the question:
Hint: if $P$ [satisfies the stated conditions], use the ideas of Ex. 4.8 to show that $\mathrm{Hom}_{\mathcal{E}}(P,-)$ preserves coproducts and hence all colimits; then use 7.13. (emphasis mine)
The referenced 7.13 roughly gives that $\phi$ is the inverse image of a geometric morphism iff it is left exact and preserves $\mathcal{S}$-indexed colimits (essentially a particular case of SAFT, although the theorem gives a third equivalent condition in terms of flat functors).
Exercise 4.8 is to show the equivalence for a geometric morphism $\gamma:\mathcal{E} \to \mathcal{S}$ of connectedness of $\mathcal{E}$, fullness of $\gamma^*$ and the preservation of colimits by $\gamma_*$. I initially supposed that it might be possible to extend this result to show that $\mathrm{Hom}_{\mathcal{E}}(P,-)$ preserving coproducts forces its left adjoint to be full, but this is in fact false whenever $P$ has non-trivial endomorphisms (ie almost always).
So how does one conclude that coequalizers are preserved?
Attempt: Given $f,g$ in $\mathcal{E}$, applying $\mathrm{Hom}_{\mathcal{E}}(P,-)$ to their coequalizer diagram: $$A \rightrightarrows B \twoheadrightarrow C,$$ where $c:B \twoheadrightarrow C$, gives $$\mathrm{Hom}_{\mathcal{E}}(P,A) \rightrightarrows \mathrm{Hom}_{\mathcal{E}}(P,B) \twoheadrightarrow \mathrm{Hom}_{\mathcal{E}}(P,C).$$ To show that the epi coequalizes $f \circ -$ and $g \circ -$ we need to construct a factoring map to an arbitrary coequalizing function. Indeed, given $h:\mathrm{Hom}_{\mathcal{E}}(P,B) \to T$ with $h(f \circ s) = h(g \circ s)$ for every $s \in \mathrm{Hom}_{\mathcal{E}}(P,A)$, the only sensible candidate for such a map is to send $c \circ t$ to $h(t)$ for each $t \in \mathrm{Hom}_{\mathcal{E}}(P,B)$. However, while this is a total function since $c \circ -$ is surjective, I can't show that it's well-defined: if $c \circ t = c \circ t'$, why should $h(t)=h(t')$? After all, there may be maps $P \to B$ which don't factor through $f$ or $g$, and we have no control over these a priori.
Side note: The question includes a citation of a French paper by J. E. Roos which I can't seem to find any scans of online; if that paper contains the answer to my question and you can provide a link to it, it would be appreciated.
Thanks to my advisor Olivia Caramello for the advice on this answer.
Since $\mathrm{Hom}_{\mathcal{E}}(P,-)$ preserves monos and epis, it preserves image factorizations. In particular, we can consider the image $R$ of $\langle f,g \rangle: A \to B \times B$, a relation on $B$ which is then mapped to a corresponding relation on $\mathrm{Hom}_{\mathcal{E}}(P,B)$. For each $n \geq 0$, $R^n$ is either a mono or is computed via repeated pullbacks and images, so $R^n$ is preserved by $\mathrm{Hom}_{\mathcal{E}}(P,-)$. The equivalence relation generated by $R$ is computed as the image of $$\coprod_{n \geq 0} R^n \to B \times B,$$ and is thus also preserved.
But the coequalizer of $f$ and $g$ is the coequalizer of the two projections of $R$, which in turn is the quotient of $B$ by the corresponding equivalence relation; this equivalence relation is then easily seen to be the kernel pair of $c$, and we have seen that this is preserved.
Phrased in another way, we have that "$c$ coequalizes $(f,g)$ if and only if its kernel pair is the equivalence relation generated by the image of $\langle f,g \rangle$", and this property is preserved by any functor preserving finite limits, epimorphisms and coproducts, so $\mathrm{Hom}_{\mathcal{E}}(P,-)$ preserves coequalizers as required.