= Solution
Let $(G,\varepsilon,\delta)$ be a <Cartesian comonad>, so its endofunctor preserves <finite limits>. A <coalgebra for a comonad> is $(A,a)$ with $a:A\to GA$, $\varepsilon_Aa=1_A$ and $\delta_Aa=Ga\,a$. Its morphisms satisfy $bf=Gf\,a$. Write $U:\mathcal E^G\to\mathcal E$ for the <forgetful functor> and $R(X)=(GX,\delta_X)$ for the <cofree coalgebra>. Transposition is
$$
\mathcal E^G((A,a),R(X))\cong\mathcal E(A,X),\qquad f\longmapsto\varepsilon_X f,\quad h\longmapsto Gh\,a.
$$
The <forgetful functor> creates <finite limits>, since $G$ preserves them. We construct both <exponential objects> and the <subobject classifier>.
For coalgebras $(A,a),(B,b)$, set $D=B^A$ and $W=(GB)^A$ in the ambient <topos>. Define $p,q:GD\to W$ by transposing
$$
p^\flat=b\,\operatorname{ev}(\varepsilon_D\times1_A),\qquad q^\flat=G\operatorname{ev}(1_{GD}\times a),
$$
where $GD\times GA\cong G(D\times A)$. Their cofree transposes are coalgebra morphisms
$$
\widehat p=Gp\,\delta_D,\quad\widehat q=Gq\,\delta_D:R(D)\rightrightarrows R(W).
$$
Let $H$ be their coalgebra <equalizer>. For a coalgebra $(C,c)$, an arrow $C\to R(D)$ corresponds to a map $h:C\times A\to B$. Its evaluation respects coalgebra structure exactly when
$$
bh=Gh\,(c\times a).
$$
Under the displayed transpositions this is precisely the condition that it equalize $\widehat p,\widehat q$. Thus $\mathcal E^G(C,H)\cong\mathcal E^G(C\times A,B)$ naturally, and $H$ is the required <exponential in a coalgebra topos>.
Let $\kappa:G\Omega\to\Omega$ classify $G\top:G1\cong1\hookrightarrow G\Omega$. In $R(\Omega)$ take the <equalizer>
$$
\Omega_G\hookrightarrow R(\Omega)\mathrel{\substack{\xrightarrow{1}\\[-2pt]\xrightarrow[G\kappa\,\delta_\Omega]{}}}R(\Omega).
$$
The truth map into $R(\Omega)$ is $G\top$ and factors through this <equalizer>. To verify its universal property, take an underlying <subobject> $m:S\hookrightarrow A$ of a coalgebra. It carries a compatible coalgebra structure exactly when $a$ maps $S$ into $GS$. Such a structure, if it exists, is unique because $Gm$ is monic; its laws follow by cancellation of $m,Gm,G^2m$. Moreover $a^{-1}(GS)\subseteq S$ always: applying $\varepsilon_A$ to a point of $GS$ lands in $S$. Thus stability is equivalent to $S=a^{-1}(GS)$.
If $\chi:A\to\Omega$ classifies $S$, this last equality is exactly
$$
\chi=\kappa G\chi\,a.
$$
The cofree transpose $\widehat\chi=G\chi\,a:A\to R(\Omega)$ satisfies this equation precisely when it factors through $\Omega_G$. Indeed transposing the equality $\widehat\chi=G\kappa\,\delta_\Omega\widehat\chi$ gives the equation above. Pulling back its truth map recovers $S$, because $\kappa$ classifies $G\top$. Hence $\Omega_G$ is the <subobject classifier of a coalgebra topos>. We have obtained <finite limits>, exponentials and a classifier, proving \b[$\mathcal E^G$ is a topos].
Now let $f:\mathcal H\to\mathcal E$ be a <geometric morphism>, and put $L=f^*$, $V=f_*$, with unit $\eta$. Both <functors> preserve <finite limits>, so $G=LV$ is a <Cartesian comonad> on $\mathcal H$. Put $\mathcal D=\mathcal H^G$. Its comparison <functor>
$$
K:\mathcal E\longrightarrow\mathcal D,\qquad K(X)=(LX,L\eta_X)
$$
is left exact and has right adjoint
$$
J(A,a)=\operatorname{Eq}\bigl(Va,\eta_{VA}:VA\rightrightarrows V L V A\bigr).
$$
To check the adjunction, transpose an underlying map $LX\to A$ to $X\to VA$. The coalgebra-morphism equation becomes exactly the equality of the two composites into $V L V A$, so the transposed arrow factors uniquely through the displayed <equalizer>.
The counit $KJ(A,a)\to(A,a)$ is invertible. Applying $L$ to the defining <equalizer> gives the <equalizer> of $Ga,\delta_A:GA\rightrightarrows G^2A$. The structure map $a:A\to GA$ is exactly that <equalizer>: it equalizes by coassociativity; if $t:T\to GA$ equalizes, apply $\varepsilon_{GA}$ to obtain $t=a\varepsilon_A t$, giving its unique factorization through $a$. Thus the underlying comparison counit is an isomorphism, and the <forgetful functor> reflects isomorphisms. Therefore $J$ is <full and faithful>.
The adjunctions $U\dashv R$ and $K\dashv J$ define
$$
\boxed{\mathcal H\xrightarrow{p}\mathcal D\xrightarrow{i}\mathcal E,\qquad f=i\circ p.}
$$
Here $p^*=U$ is faithful, so $p$ is a <surjective geometric morphism>; $i_*=J$ is full and faithful, so $i$ is a <geometric embedding>. Their inverse-image composite is $UK=L$, establishing the <surjection-embedding factorization of a geometric morphism>.
For uniqueness, let $S$ be the class of arrows in $\mathcal E$ inverted by $f^*$. In any surjection-embedding factorization $f=i'p'$, conservativity of $p'^*$ shows that $S$ is also the class inverted by $i'^*$. The essential image of the fully faithful $i'_*$ consists exactly of the $S$-local objects $Y$, namely those for which $\mathcal E(B,Y)\to\mathcal E(A,Y)$ is bijective for every $A\to B$ in $S$. One direction follows from the adjunction. For the converse, the reflector unit $Y\to i'_*i'^*Y$ belongs to $S$; locality gives a retraction, and locality of the reflected object makes that retraction also a right inverse. Thus the unit is invertible. The image subcategory is consequently determined by $f$, independently of the factorization. The two intermediate toposes are equivalent over $\mathcal E$, and their inverse-image composites with the embeddings determine the surjection legs. This proves \b[uniqueness up to equivalence].
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