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Take the definition of an elementary topos as a category with finite limits, exponential objects and a subobject classifier $\top :1\hookrightarrow \Omega$. Its power object is $PX={\Omega }^X$, and $Pf:PY\to PX$ is precomposition with $f:X\to Y$. Transposing a predicate on $X\times Y$ in either variable givesThus $P^{\mathrm{op}}:\mathcal{E}\to {\mathcal{E}}^{\mathrm{op}}$ is left adjoint to the contravariant power-object functor $P:{\mathcal{E}}^{\mathrm{op}}\to \mathcal{E}$.

Here are the remaining hypotheses for the crude monadicity theorem, obtained from those topos axioms. The unit of this adjunction is ${\eta }_X:X\to PPX$, internally ${\eta }_X(x)(S)=(x\in S)$. It is monic: if two such evaluations agree, evaluate on the singleton predicate $\{x\}$ classified by the diagonal to obtain $x=y$. If $Pf$ is invertible, naturality $PPf{\eta }_X={\eta }_{Y}f$ therefore makes $f$ monic. The characteristic map of this mono, regarded as a global element of $PY$, pulls back along $f$ to the everywhere-true predicate on $X$. Since $Pf$ is monic, that characteristic map was already everywhere true on $Y$. The classifier pullback then says that $f$ is invertible. Hence $P$ is a conservative functor.

It remains to check preservation of reflexive coequalizers in the opposite category. Equivalently, let $f,g:B\rightrightarrows A$ be a coreflexive pair, with $r:A\to B$ satisfying $rf=rg=1_B$, and let $e:E\hookrightarrow B$ be their equalizer. Both $f$ and $g$ are monic. For any mono $i$, there is a direct-image map ${\exists }_i$ between power objects: a subobject is sent to its composite with $i$. This uses only classification of monos, not the prior existence of general images or colimits. We have $Pi{\exists }_i=1$.

For a predicate $W\hookrightarrow B$, consider its direct image along $f$. Its pullbacks along the two sections areTo verify the second equality, $g(b)=f(w)$ with $w\in W$ implies $b=w$ after applying $r$, and therefore $f(b)=g(b)$; conversely $b\in W\cap E$ supplies that witness. This argument works for parameterized subobjects as well, so it is an equality of arrows between power objects.

Now if $h:PB\to Z$ satisfies $hPf=hPg$, then $h=h{\exists }_{e}Pe$. Hence $h{\exists }_e:PE\to Z$ is its unique factorization through $Pe$, uniqueness following from the section ${\exists }_e$. Thereforeis a coequalizer. Finite limits supply every required coreflexive equalizer. The right adjoint $P$ reflects isomorphisms and preserves the corresponding reflexive coequalizers, so $P$ is monadic. In particular ${\mathcal{E}}^{\mathrm{op}}$ is equivalent to the Eilenberg-Moore category of the double-power-object monad on $\mathcal{E}$.

Let $F:\mathcal{E}\to \mathcal{F}$ be a logical functor, and suppose $L⊣F$. Preservation of exponentials and the classifier gives $F{P}_{\mathcal{E}}\cong P_{\mathcal{F}}{F}^{\mathrm{op}}$, compatibly with the units, counits and resulting monads. Thus $F^{\mathrm{op}}$ is the algebra functor lifting the base functor $F$ through the two monadic power-object functors. The adjoint lifting theorem for monad algebra functors applies to the base adjunction $L⊣F$: its required coequalizers exist in ${\mathcal{E}}^{\mathrm{op}}$, because $\mathcal{E}$ has finite equalizers. It supplies a left adjoint $K⊣F^{\mathrm{op}}$. Taking opposites gives $F⊣K^{\mathrm{op}}$, the required right adjoint to $F$.

Preserving either of the two logical structures by itself is insufficient. For the exponential example take $F:\mathbf{Set}\to \mathbf{Set}$ constant at $1$. The constant-empty functor is its left adjoint, since both relevant hom-sets are singletons. Its canonical exponential comparison is $1\cong 1^1$, so it preserves exponentials. It has no right adjoint: a functor with a right adjoint would preserve the initial object, whereas $F(\varnothing )=1$.

For the classifier example take $G:(\mathbb{Z}/2)\text{-}\mathbf{Set}\to \mathbf{Set}$ to be fixed points. The trivial-action functor is left adjoint to $G$. In a group-action topos the classifier is the trivial-action two-element set, since invariant subsets have ordinary equivariant characteristic maps. Therefore $G$ preserves the classifier, its true arrow and the terminal object. But $G$ does not preserve the coequalizer of the identity and the nontrivial translation on the regular two-element group set: that coequalizer is $1$, while the fixed-point sets of the domain and codomain of the parallel pair are empty. Their set-theoretic coequalizer is empty, not $G(1)=1$. Thus $G$ has no right adjoint.

Write the Cartesian comonad as $(G,\epsilon ,\delta )$, with $G$ preserving finite limits. A coalgebra for a comonad is $(A,a)$ with $a:A\to GA$, ${\epsilon }_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. Its forgetful functor $U:{\mathcal{E}}^G\to \mathcal{E}$ is faithful, creates finite limits, and has the cofree coalgebra right adjoint $R(X)=(GX,{\delta }_X)$. We construct the two remaining topos structures explicitly.

For exponential objects, take coalgebras $(A,a),(B,b)$ and start with $R(B^A)$. Put $X=G(B^A)$, with structure $\xi ={\delta }_{B^A}$, and letTranspose the following two maps $X\times A\to GB$ into maps $u,v:X\to (GB{)}^A$:using $G(X\times A)\cong GX\times GA$. The cofree adjunction transposes $u,v$ once more into coalgebra morphisms $R(B^A)\rightrightarrows R((GB{)}^A)$. Let $Q$ be their equalizer. A coalgebra map $(C,c)\to R(B^A)$ corresponds to an arbitrary ambient map $h:C\times A\to B$. It factors through $Q$ precisely whenwhich is exactly the condition that $h$ be a coalgebra morphism. Therefore $Q$ represents ${\mathcal{E}}^G(C\times A,B)$ and is the required exponential in a coalgebra topos.

For the subobject classifier, let $\kappa :G\Omega \to \Omega$ classify the mono $G\top :G1\cong 1\hookrightarrow G\Omega$. In the cofree coalgebra $R\Omega$, formBoth arrows are coalgebra morphisms. The cofree transpose of $\top :1\to \Omega$ factors through this equalizer and gives its true arrow. To verify classification, let $S\hookrightarrow X$ have ambient characteristic map $\chi :X\to \Omega$. It supports a subcoalgebra of $(X,x)$ exactly when it is invariant under $x$, equivalentlyThe counit proves the reverse containment in the first equation; the forward containment supplies the restricted structure map, whose coalgebra laws follow through the mono. Under the cofree adjunction, the second equation says exactly that $G\chi x:X\to R\Omega$ factors through ${\Omega }_G$. Pulling back its true arrow recovers $S$, since $\kappa G\chi x=\chi$. This proves the universal property of the subobject classifier of a coalgebra topos. Hence ${\mathcal{E}}^G$ is a topos.

Now let $f:\mathcal{E}\to \mathcal{F}$ be a geometric morphism, with $L=f^{\ast }⊣H=f_{\ast }$. The comonad $G=LH$ is Cartesian: $L$ preserves finite limits and the right adjoint $H$ preserves limits. Put $\mathcal{D}={\mathcal{E}}^G$, already a topos. The forgetful adjunction $U⊣R$ defines $p:\mathcal{E}\to \mathcal{D}$ with $p^{\ast }=U$, so $p^{\ast }$ is faithful.

The comparison functorpreserves finite limits. It has a right adjoint $J$, given on a coalgebra $(A,a)$ byIndeed, the transposed arrow $Y\to HA$ corresponds to a coalgebra map $KY\to (A,a)$ exactly when it equalizes these two maps. Applying the finite-limit-preserving $L$ shows that $LJ(A,a)$ is the equalizer ofThat equalizer is $a:A\to GA$. If $t:Z\to GA$ equalizes the pair, then $a{\epsilon }_{A}t={\epsilon }_{GA}Gat={\epsilon }_{GA}{\delta }_{A}t=t$, proving the claimed universal property. Consequently the counit $KJ(A,a)\to (A,a)$ is invertible. The fully faithful adjoint criterion makes $J$ full and faithful.

Thus $K⊣J$ defines a geometric embedding $i:\mathcal{D}\to \mathcal{F}$, and $UK=L$ identifies the composite with $f$. The requested factorization iswhere $p$ is a surjective geometric morphism and $i_{\ast }=J$ is full and faithful.

Write the comonad as $(G,ϵ,\delta )$ and its category of coalgebras for a comonad as ${\mathcal{E}}^G$. A coalgebra for a comonad is a map $a:A\to GA$ with ${ϵ}_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. A morphism $f:(A,a)\to (B,b)$ satisfies $bf=Gfa$. Let $U:{\mathcal{E}}^G\to \mathcal{E}$ be the forgetful functor and let $R(X)=(GX,{\delta }_X)$ be the cofree coalgebra. The adjunction $U⊣R$ has the explicit correspondenceWe construct the three pieces of the elementary topos structure.

Because $G$ preserves finite limits, each underlying finite limiting cone has a unique coalgebra structure induced by the structures on its vertices. The counit and coassociativity equations can be checked after its jointly monic projections. Thus $U$ creates finite limits and reflects isomorphisms. A morphism of coalgebras is monic exactly when its underlying morphism is monic, by the diagonal criterion using the created pullback.

For exponentials in a coalgebra topos, fix coalgebras $(A,a)$ and $(B,b)$ and put $E=B^A$ in $\mathcal{E}$. On the cofree coalgebra $R(E)$ there is an underlying evaluationThe two mapsuse $G(GE\times A)\cong GGE\times GA$ in the second expression. Transpose them in $\mathcal{E}$ to maps $r,s:GE\rightrightarrows (GB{)}^A$, and then transpose across $U⊣R$ to coalgebra morphisms $\tilde{r},\tilde{s}:R(E)\rightrightarrows R((GB{)}^A)$. Take their equalizer $C$ in ${\mathcal{E}}^G$.

An underlying map $UX\times A\to B$ corresponds to a coalgebra map $h:X\to R(E)$. The equation saying that the original map is a coalgebra morphism is precisely $rh=sh$, since ${\delta }_{E}h=Ghx$ for the structure $x:X\to GX$. By the cofree adjunction, this is equivalent to $\tilde{r}h=\tilde{s}h$, hence to unique factorization through $C$. Thereforenaturally in $X$. This constructs the required exponential object.

For the subobject classifier of a coalgebra topos, let $\top :1\hookrightarrow \Omega$ be the underlying subobject classifier, and let $\kappa :G\Omega \to \Omega$ classify the mono $G\top :G1\cong 1\hookrightarrow G\Omega$. Its cofree transpose is the coalgebra endomorphismDefine ${\Omega }_G$ as the equalizer of $k$ and $1_{R\Omega }$. The transpose of $\top$ factors through this equalizer and gives ${\top }_G:1\to {\Omega }_G$.

Indeed, for a subobject $S\hookrightarrow UX$ classified by $\chi :UX\to \Omega$, the pullback $x^{-1}(GS)$ has characteristic map $\kappa G\chi x$. It is always contained in $S$, by naturality of the counit. Equality holds exactly when $x$ restricts to a coalgebra structure on $S$; its axioms then follow by composing with the monomorphisms $GS\hookrightarrow GX$ and $GGS\hookrightarrow GGX$. Under $U⊣R$, the classifying map becomes $h=G\chi x:X\to R\Omega$. The equality of subobjects is ${ϵ}_{\Omega }h=\kappa h$. Transposing this equality gives $h=kh$, so exactly the coalgebra subobjects correspond to maps $X\to {\Omega }_G$. Their pullback of ${\top }_G$ is the desired subcoalgebra, and uniqueness follows from uniqueness of $\chi$.

Thus we have finite limits, exponentials and a subobject classifier:The construction does not assume that $G$ preserves the underlying exponentials or underlying subobject classifier.

A Cartesian closed category has finite products in a category and, for every $F$, an exponential object $G^F$ characterized naturally byIn $[\mathcal{C},\mathbf{Set}]$, finite products and the terminal functor are computed pointwise. With $h_B=\mathcal{C}(B,-)$, defineThis is a set because $\mathcal{C}$ is a small category. For $v:B\to B^{\prime }$, precomposition $h_{B^{\prime }}\to h_B$ induces $(G^F)(v)$ by precomposition of natural transformations, making $G^F$ a covariant functor.

Given $\alpha :K\times F\to G$, define its curry $\hat{\alpha }:K\to G^F$ byFor $a:C\to C^{\prime }$, naturality of $\alpha$ gives $G(a){\alpha }_C(K(u)y,x)={\alpha }_{C^{\prime }}(K(au)y,F(a)x)$, proving that this is a natural transformation $h_B\times F\to G$. The same formula with $u$ precomposed proves naturality of $\hat{\alpha }$ in $B$.

Conversely, from $\beta :K\to G^F$ defineNaturality of $\beta$ and its values proves naturality of $\alpha$. Currying this result recovers $\beta$, since ${\beta }_C(K(u)y)=(G^F)(u)({\beta }_B(y))$; uncurrying a curry recovers $\alpha$ by setting $u=1_B$. These mutually inverse constructions are natural in $K,G$, giving the adjunction required for an exponential object. Therefore

Suppose the small category $\mathcal{C}$ is Cartesian closed. The Yoneda embedding preserves finite products. For $A,B,X\in \mathcal{C}$, the Yoneda lemma and the exponential object adjunction giveThe bijections are natural in $X$, soThus the essential image of $\mathcal{C}$ in its presheaf topos is full and closed under finite products and exponentials.

Let $B$ be a subterminal object in a Cartesian closed category. For every $X$ and $A$, the exponential object adjunction givesThe set on the right has at most one element because $B$ is subterminal. Hence $B^A$ is subterminal, proving that $\mathrm{Sub}(1)$ is an exponential ideal.

Now let $\mathcal{D}\subseteq \mathcal{C}$ be reflective, with reflector $L$ and unit ${\eta }_A:A\to LA$. Recall the useful form of its universal property: an object $D$ lies in $\mathcal{D}$ precisely when every map $f:A\to D$ factors uniquely through ${\eta }_A$.

Suppose first that $\mathcal{D}$ is an exponential ideal. Reflective subcategories are closed under ambient limits, so $LA\times LB$ lies in $\mathcal{D}$. Given $f:A\times B\to D$ with $D\in \mathcal{D}$, curry in the first variable to obtain $A\to D^B$. Since $D^B\in \mathcal{D}$, this factors uniquely through ${\eta }_A$ and uncurries to $LA\times B\to D$. Curry once more, now in $B$; since $D^{LA}\in \mathcal{D}$, the result factors uniquely through ${\eta }_B$. Thus every $f$ factors uniquely throughThis makes $LA\times LB$ a reflection of $A\times B$, so uniqueness of reflections gives

Conversely, suppose $L$ preserves binary products, and take $D\in \mathcal{D}$. To prove $D^A\in \mathcal{D}$, start with $f:X\to D^A$ and let $\tilde{f}:X\times A\to D$ be its transpose. Since $D$ is reflective, $\tilde{f}$ factors uniquely throughComposing the resulting map $LX\times LA\to D$ with $1_{LX}\times {\eta }_A$ and currying produces an extension $LX\to D^A$ of $f$. Product preservation identifies the unit on $LX\times A$ with $1_{LX}\times {\eta }_A$, so the same universal property proves uniqueness. Therefore $D^A$ is reflective, and $\mathcal{D}$ is an exponential ideal. This proves the reflector product criterion for an exponential ideal.

Finally consider the arrow category $[2,\mathbf{Set}]$. For arrows $a:A_0\to A_1$ and $b:B_0\to B_1$, letThen the exponential $b^a$ is the arrowIndeed, a commutative square from $c:C_0\to C_1$ to this arrow is, after currying, exactly a commutative square $a\times c\to b$. This establishes the required exponential adjunction and proves that $[2,\mathbf{Set}]$ is cartesian closed.

If $b$ is injective, two elements $(u_0,u_1)$ and $(u_0^{\prime },u_1)$ of $F$ haveso injectivity gives $u_0=u_0^{\prime }$. Hence $F\to B_1^{A_1}$ is injective. The category of injective functions is therefore an exponential ideal in the arrow category. Its terminal object and binary products are inherited pointwise, so these same exponential objects make $\mathrm{Mono}(\mathbf{Set})$ cartesian closed.

If every slice category $\mathcal{C}/A$ is essentially small, the displayed exponential object and the set of sieves on a category at each $A$ are small even when $\mathcal{C}$ is large. Together with pointwise finite limits, they give the elementary-topos constructions in a universe admitting the large diagrams. This does not force the entire functor category to be locally small in the original universe: a large discrete category has terminal slices, but maps between its constant-one and constant-two presheaves are arbitrary class-indexed binary families.

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.