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Here Cartesian category means a category with all finite limits. Let $m:Y\hookrightarrow X$ be a monomorphism, and let ${\epsilon }_X\hookrightarrow X\times PX$ be universal membership for a power object $PX$. Pull it back to $Y\times PX$ and then regard the result as a subobject of $X\times PX$. Its unique classifier is an endomorphism $r:PX\to PX$: on a parameterized subobject it intersects with $Y$. Intersecting twice has the same result as intersecting once, so uniqueness of classification gives $r^2=r$.

Take the equalizer $q:Q\hookrightarrow PX$ of $r$ and the identity, and restrict membership to $Y\times Q$. For any subobject $S\hookrightarrow Y\times Z$, its composite with $m\times 1_Z$ is a subobject of $X\times Z$, hence is uniquely classified by $s:Z\to PX$. Since it is contained in $Y\times Z$, its intersection with that subobject changes nothing, so $rs=s$ and $s$ factors uniquely through $Q$. Conversely every map $Z\to Q$ gives the original kind of family by pulling back the restricted membership. These inverse operations givenaturally in $Z$. Thus $Q$ is a power object of a subobject, even if no power objects other than $PX$ were assumed.

Now consider the slice category $\mathcal{E}/A$, which has finite limits formed by ambient pullbacks and has terminal object $1_A:A\to A$. The constant family $A\times X\to A$ has slice power object $A\times PX\to A$: for $Z\to A$,and a classifier $Z\to PX$ is equivalently the map $Z\to A\times PX$ over $A$. Every family $p:X\to A$ is a subobject of this constant family through its monic graph $(p,1_X):X\hookrightarrow A\times X$. Apply the first construction inside the slice. Consequently every slice of a Cartesian category with power objects again has power objects.

Take two subobjects $A,B\hookrightarrow X$ whose inverse images coincide. Their characteristic maps $a,b:X\to {\Omega }_{\mathcal{E}}$ satisfy $\bar{\lambda }{f}^{\ast }a=\bar{\lambda }{f}^{\ast }b$ by part (i). Under the adjunction $f^{\ast }⊣f_{\ast }$, these maps transpose to $\lambda a$ and $\lambda b$, respectively, by naturality of the unit. If $\lambda$ is a monomorphism, then $a=b$ and hence $A=B$. Thus $f^{\ast }$ reflects equality of subobjects.

For parallel arrows $r,s:X\rightrightarrows Y$ with $f^{\ast }r=f^{\ast }s$, their equalizer $K\hookrightarrow X$ pulls back to all of $f^{\ast }X$. Reflection of equality with the whole subobject gives $K=X$, so $r=s$. Therefore the inverse image is faithful, and the geometric morphism is surjective. This is the surjectivity detected by the subobject classifier criterion. For completeness, faithfulness is equivalent here to conservativity: if $f^{\ast }h$ is invertible, the image inclusion of $h$ and its kernel-pair diagonal become whole subobjects; reflection makes $h$ both a regular epimorphism and monic, hence invertible in the topos.

For a subobject $m:S\hookrightarrow X$, define the closure operation induced by a left-exact reflector bywhere the pullback uses ${\eta }_X:X\to LX$ and the monomorphism $Lm:LS\hookrightarrow LX$. Naturality of the unit factors $S\hookrightarrow X$ through $c_X(S)$, proving extensivity, and functoriality of $L$ proves monotonicity. Applying the left-exact $L$ to the defining pullback gives $L(c_X(S))\cong LS$, since $L{\eta }_X$ is invertible. Repeating the construction therefore changes nothing, proving idempotence.

For $h:Y\to X$, naturality of $\eta$ and preservation of pullbacks identify the closure of $h^{\ast }S$ with $Y{\times }_{LX}LS$, which is also $h^{\ast }{c}_X(S)$. HenceThe operation is thus universal: it is stable under every pullback. The same construction preserves finite intersections because $L$ is left exact.

Every characteristic map $a:X\to \Omega$ expresses its subobject as $a^{\ast }\top$. Universality of the closure operation induced by a left-exact reflector givesBy definition, $j$ classifies $c_{\Omega }(\top )$, so the right-hand subobject has characteristic map $j\circ a$. The original and its closure agree exactly when their unique characteristic maps agree. ThusExtensivity, idempotence, preservation of intersections and closure of the whole subobject also give the Lawvere-Tierney topology laws for $j$.

Let ${\Omega }_j\hookrightarrow \Omega$ be the equalizer of $j$ and the identity. By part (iv), maps $X\to {\Omega }_j$ correspond exactly to closed subobjects of $X$. Closed subobjects of $X$ in turn correspond to subobjects of $LX$ in $\mathcal{L}$: send $S$ to $LS\hookrightarrow LX$, and in the reverse direction pull a fixed subobject $N\hookrightarrow LX$ back along ${\eta }_X$. Left exactness shows that these operations are inverse; the pulled-back object has reflection $N$, making it closed by the defining formula.

ConsequentlyReplacing $X$ by $LX$ and using $L{\eta }_X$ invertible shows that maps into ${\Omega }_j$ extend uniquely across every unit. The criterion of part (i) therefore proves ${\Omega }_j$ itself is fixed, so belongs to $\mathcal{L}$. Truth factors through it because $j\top =\top$. For fixed $X$, the displayed bijection becomesWith the factored truth arrow this is precisely the subobject classifier universal property. Thus the closed-subobject classifier supplies the classifier in the reflective subcategory.

Interpret the given relation by a subobject $R\hookrightarrow A\times B$ and write its projections as $p:R\to A$, $q:R\to B$. Totality says that the image of $p$ is all of $A$, so $p$ is a regular epimorphism. Single-valuedness makes $p$ monic: if $pr=ps$ for two arrows into $R$, the functionality sequent forces $qr=qs$, and the inclusion into $A\times B$ then gives $r=s$.

A regular epimorphism that is monic is invertible. Indeed its kernel pair consists of equal projections, and its coequalizer universal property supplies an inverse. Therefore defineThe graph of this map is exactly the original relation $R$, since $(p,q)=(1_A,f)p$. The graph also makes the map unique. This proves total functional relations define morphisms, using regular-category semantics and no choice of individual witnesses.

A coherent category is a regular category with finite unions of subobjects, including the empty union, stable under pullback. Thus it has finite limits, pullback-stable regular-epimorphism/monomorphism image factorizations, and pullback-stable finite joins in each subobject lattice.

Write $U=A\vee B\hookrightarrow X$ and $C=A\wedge B=A{\times }_{X}B$. Their inclusions give a commutative squareTo establish the pushout property, take arrows $a:A\to Y$, $b:B\to Y$ agreeing on $C$. Regard their graphs as subobjects of $U\times Y$ and form their union $R$. Internally, this relation says that a point of $U$ is related to its $a$-value when it lies in $A$, or to its $b$-value when it lies in $B$.

The relation is total because every point of $U$ belongs to at least one of $A,B$. It is single-valued: two witnesses in the same summand give the same value by functionality of that map; witnesses in different summands have their shared point in $C$, where the two maps agree. By the relation argument proved in Question 7's unheaded continuation, the projection $R\to U$ is both a regular epimorphism and monic, hence an isomorphism. It consequently gives a map $h:U\to Y$ restricting to $a,b$.

For uniqueness, the equalizer of two possible maps contains $A$ and $B$, hence contains their union $U$ and is all of $U$. Thus the square is a pushout, provingThis is the unions of subobjects are pushouts property; it does not assume arbitrary pushouts already exist in a coherent category.

Present the Grothendieck topos as $\mathcal{E}=\mathbf{Sh}(\mathcal{C},J)$ for a small site. Let $a:\hat{\mathcal{C}}\to \mathcal{E}$ be sheafification and $i$ the inclusion. The standard sheafification construction supplies $a⊣i$, with $a$ preserving finite limits. Concretely, matching-family constructions commute with finite limits, and their directed refinement over covering sieves does so as well. Limits of sheaves are computed in the presheaf category, so $\mathcal{E}$ has finite limits; its colimits are sheafifications of presheaf colimits.

For sheaves $A,B$, form the presheaf exponentialIt is a sheaf. Indeed, for a covering sieve $S\hookrightarrow yc$, one has $aS\cong a(yc)$. Exponential adjunction, sheafification adjunction and left exactness giveThese are the restriction comparisons, proving the sheaf condition. Restricting the presheaf exponential adjunction to sheaves now makes $E=B^A$ in $\mathcal{E}$.

The subobject classifier is the sheaf ${\Omega }_J$ of J-closed sieves. A sieve $S$ on $c$ is J-closed when, for every $f:d\to c$, $f^{\ast }S\in J(d)$ implies $f\in S$. Pullback of sieves defines its restrictions, and the maximal sieve defines truth. Local sieve data glue by taking the J-closure of their compatible generated sieve; uniqueness follows because membership in a J-closed sieve is local. This proves that ${\Omega }_J$ is a sheaf.

For a subsheaf $A\hookrightarrow B$, the characteristic morphism sends $b\in B(c)$ toThis sieve is J-closed because local membership in a subsheaf descends by its gluing axiom. Pulling truth back recovers exactly $A$. Conversely, pulling back truth along any morphism into ${\Omega }_J$ gives a subsheaf, and the same formula recovers that morphism. Finite limits, exponentials and this classifier make every Grothendieck topos an elementary topos.

Here are the explicit Heyting operations on subsheaves of an object $B$. Intersections give meets:The top is $B$. Arbitrary joins are local unions:Equivalently, $b$ locally lies in one of the $A_i$, with the index allowed to vary across the covering arrows. This is the J-closure of the pointwise union, or its sheafification. It is the smallest subsheaf containing every $A_i$, since any such subsheaf must contain sections locally in it.

The bottom is the empty join, namely the initial subobject. Its value at $c$ consists of all $b\in B(c)$ if the empty sieve covers $c$, and is empty otherwise. This detail matters on sites with empty covers; it is not safe to use an always-empty presheaf as the initial sheaf.

Implication is described without any pointwise-complement assumption:Restrictions preserve this condition. It is also local: pull a covering sieve back along an arbitrary $f$, use membership in $A$ on those restrictions, then descend their membership in $D$. Thus it is a subsheaf. It satisfies the defining Heyting algebra adjunctionFor the forward direction take the identity restriction. For the reverse direction every restriction of a section of $C$ remains in $C$, so membership in $A$ forces membership in $D$. Negation is $\neg A=A\Rightarrow 0$. These formulas give the complete Heyting algebra structure on ${\mathrm{Sub}}_{\mathcal{E}}(B)$; in general negation is a pseudocomplement rather than a set-theoretic complement.

A geometric formula is built from atomic formulas, including equality, using finite conjunctions, arbitrary set-indexed disjunctions and existential quantification in a finite variable context. Truth is the empty conjunction and falsity the empty disjunction. A geometric theory is a set of sequents $\varphi {⊢}_{\vec{x}}\psi$ between such formulas over a many-sorted signature. The context supplies the universal force of an axiom; unrestricted universal quantification, implication and negation are not formula constructors in this fragment.

These choices are exactly suited to inverse image functors of geometric morphisms. Finite-limit preservation handles equality and conjunction; colimit and image preservation handle disjunction and existential quantification. Therefore an inverse image carries an internal model of a geometric theory to another model.

Construct the geometric syntactic category ${\mathcal{C}}_{\mathbb{T}}$ as follows. Objects are formulas in context $\{\vec{x}.\varphi \}$, modulo provable renaming and equivalence. A morphism to $\{\vec{y}.\psi \}$ is an equivalence class of formulas $\theta (\vec{x},\vec{y})$ that are provably total and single-valued:The last notation abbreviates componentwise equality. Identity is equality of the context variables; composition is existential conjunction over the intermediate tuple. The category has finite limits, formed through conjunction and equality.

Give it the geometric syntactic topology $J_{\mathbb{T}}$. A family of arrows represented by ${\theta }_i({\vec{y}}_i,\vec{x})$ into $\{\vec{x}.\varphi \}$ covers whenPullback stability is substitution, and transitivity comes from distributing existential conjunction through the covering disjunctions. Thus the generated sieves define a Grothendieck topology. Empty covers impose the interpretation of falsity as the initial object.

The classifying topos isIts universal model $U_{\mathbb{T}}$ assigns a sort $A$ the sheafified representable of $\{x^A.\top \}$, functions their definable graph morphisms and relations their definable subobjects. The syntactic covering conditions make the axioms valid in this model.

For the universal property, interpreting formulas in a model $M$ in a Grothendieck topos $\mathcal{F}$ gives a finite-limit-preserving, $J_{\mathbb{T}}$-continuous functor ${\mathcal{C}}_{\mathbb{T}}\to \mathcal{F}$: covers go to jointly epimorphic families precisely because the relevant sequents hold. The Diaconescu equivalence for geometric morphisms associates to this functor a geometric morphism $\mathcal{F}\to \mathbf{Set}[\mathbb{T}]$. Conversely, pulling $U_{\mathbb{T}}$ back along a geometric morphism gives a model. The constructions on models and their homomorphisms are inverse up to natural isomorphism, yieldingnaturally in $\mathcal{F}$. This proves the required classifying property, not merely classification of set-valued models.

The limit form of the Special adjoint functor theorem is as follows. Let $\mathcal{C}$ be a locally small category, a complete category, and a well-powered category, and suppose it has a small cogenerating family $(Q_i{)}_{i\in I}$. Let $\mathcal{D}$ be a locally small category. Then a functor $G:\mathcal{C}\to \mathcal{D}$ has a left adjoint if and only if it preserves all small categorical limits. The dual exchanges completeness, well-poweredness and cogenerators for cocompleteness, well-copoweredness and generators, and characterizes functors with a right adjoint.

Here a cogenerating set distinguishes unequal parallel maps by postcomposition into one of the $Q_i$. A well-powered category has a set of subobjects of each fixed object. Completeness means existence of all small categorical limits. We use the initial-object lemma for complete categories with a weakly initial set: a complete category that is locally small has an initial object exactly when it has a weakly initial set. A proof is given in Question 5(c). We also use the fact that when $G$ preserves small categorical limits, the comma category $(B\downarrow G)$ is complete, with its limits constructed in $\mathcal{C}$; local smallness is inherited from $\mathcal{C}$. Finally, having an initial object in $(B\downarrow G)$ for each $B$ is the universal arrow from an object to a functor characterization of a left adjoint.

Assume $G$ preserves small categorical limits, and fix $B\in \mathcal{D}$. We will construct a weakly initial set in $(B\downarrow G)$, rather than assume a solution set. Start with an arbitrary object $(C,x)$, where $x:B\to GC$. Call a subobject $m:S\hookrightarrow C$ supporting if $x=G(m)x_S$ for some $x_S:B\to GS$. There is at least one, namely $1_C$. Because $G$ preserves pullbacks in a category, it preserves monomorphisms: the self-pullback characterization of a monic map is preserved. Consequently each $x_S$ is unique.

There is a set of supporting subobjects by well-poweredness. Their intersection of subobjects exists by completeness; write it $m_0:C_0\hookrightarrow C$. Applying $G$ to this intersection categorical limit gives a unique $x_0:B\to G{C}_0$ lifting all the $x_S$, and hence $x=G(m_0)x_0$. Minimality says that any subobject $n:R\hookrightarrow C_0$ supporting $x_0$ is invertible: $m_{0}n$ supports $x$, so $m_0$ factors through $m_{0}n$, giving a right inverse to the monic $n$.

If $u,v:C_0\to Q_i$ satisfy $G(u)x_0=G(v)x_0$, their equalizer supports $x_0$, since $G$ preserves that equalizer. It is therefore an isomorphism, and $u=v$. We have obtained an injective functionLet $J_i$ be its image, a subset of the fixed set $S_i=\mathcal{D}(B,G{Q}_i)$. The evaluation embedding into cogenerator products is a monomorphismIndeed each $s\in J_i$ corresponds to a unique map $C_0\to Q_i$, and the cogenerating set property makes the resulting family jointly monic. Notice that we use the realized subsets $J_i$, not all of $S_i$: there need not be a map $C_0\to Q_i$ available for an unused index.

There is a set of tuples $J=(J_i)$, since $I$ and every $S_i$ are sets. For each resulting $P_J$, choose representatives of its subobjects; these form a set by well-poweredness. For every representative $R$, include every map $y:B\to GR$, a set by local smallness of $\mathcal{D}$. All resulting pairs $(R,y)$ form a set. Our given $(C,x)$ receives a map from such a pair: transport $x_0$ along the isomorphism between $C_0$ and the chosen representative, then compose its inverse with $m_0$. Thus these pairs form a weakly initial set in $(B\downarrow G)$. This is the cogenerator bound for comma-category solution sets.

The initial-object lemma for complete categories with a weakly initial set gives an initial object $(LB,{\eta }_B)$. For $h:B\to B^{\prime }$, define $Lh$ by the unique equationUniqueness proves the functor laws and gives the natural bijectionsThus $L⊣G$. Conversely, a right adjoint preserves small categorical limits: its adjunction identifies maps into the proposed limiting object with compatible families of maps into the diagram. This proves both directions of the limit form of the special adjoint functor theorem. Applying the argument to the opposite categories proves the dual statement. Choices of representatives and adjoint objects are understood in the usual ambient-universe convention for large categories.

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.

A Cartesian theory has a many-sorted first-order signature, with finite arities, and axioms expressed as sequents between Cartesian formulas. Those formulas use atomic relations and equality, truth and finite logical conjunctions. They also admit existential quantification when the quantified witness is provably unique: to form $\exists y\varphi (x,y)$, require $\varphi (x,y)\wedge \varphi (x,y^{\prime })⊢y=y^{\prime }$ relative to the theory. One can quantify a finite tuple in the same way. Arbitrary existential quantifiers, disjunction, negation and universal quantifiers are not formula constructors in this fragment. In particular falsity is not included as an additional finite-limit constructor. The universal force of a sequent comes from its free-variable context. This is the fragment whose categorical semantics requires exactly finite limits.

The Cartesian syntactic category ${\mathcal{C}}_{\mathbb{T}}$ has objects $\{\vec{x}\mid \varphi \}$, identifying harmless renamings and provably equivalent formulas-in-context. An arrow from $\{\vec{x}\mid \varphi \}$ to $\{\vec{y}\mid \psi \}$ is represented by a provably functional Cartesian relation $\theta (\vec{x},\vec{y})$: it entails $\varphi \wedge \psi$, is total on $\varphi$, and has a unique output tuple. Provable equivalence identifies arrows. Equality supplies identity arrows; composition quantifies the uniquely determined intermediate tuple in a conjunction. Totality and uniqueness prove that the composite is again functional.

The empty truth context is terminal. Products concatenate disjoint contexts and conjoin their formulas. A pullback of two functional relations adds the condition that their outputs agree; the common output is unique, so the existential quantification used to express it is Cartesian. This gives all finite limits and makes the usual equality diagrams into equalizers.

An internal model $M$ in a finite-limit category $\mathcal{E}$ interprets each sort by an object, functions by arrows and relations by subobjects. Equality is a diagonal and conjunction is a pullback intersection. A uniquely witnessed relation projects monomorphically onto the context, so its existential interpretation needs no general image operation: it is that mono. Evaluating formulas and functional relations yields a finite-limit-preserving functor $F_M:{\mathcal{C}}_{\mathbb{T}}\to \mathcal{E}$.

Conversely, the canonical syntactic model uses the single-sort truth contexts, term graphs and atomic subobjects. Its interpretation of a formula is the formula-in-context object itself. Applying any finite-limit-preserving functor supplies a model and preserves its axioms. These constructions are inverse up to the natural interpretation isomorphisms; model homomorphisms correspond to natural transformations. Therefore

For completeness, let a sequent in context be represented by subobjects $i:A\hookrightarrow C$ and $j:B\hookrightarrow C$ for its antecedent and consequent. Derivability is exactly the existence of a factorization $i=jv$ in the syntactic category. Consider the covariant representable functor ${\mathcal{C}}_{\mathbb{T}}(A,-)$, which preserves all existing limits, and hence corresponds to a set-valued model. In that model the element $i\in {\mathcal{C}}_{\mathbb{T}}(A,C)$ belongs to the antecedent, because it is the image of $1_A$. If the sequent holds, it also belongs to the consequent, so $i=jv$ for some $v:A\to B$. That is precisely a derivation of the sequent. Thus validity in all set-valued models implies Cartesian derivability. More concretely, every underivable sequent is refuted by this representable model. No assumption that every sort is inhabited is needed: some hom-sets may be empty.