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In the finite-limit convention, a Cartesian category is a category with all finite limits, equivalently a terminal object and all pullbacks in a category. Some authors use the term for finite products alone, so the convention matters in constructions requiring equalizers.

Given a finite-product category and a product-closed upward filter $\Phi$ of subobjects of its terminal object, a morphism $A\to B$ in ${\mathcal{C}}_{\Phi }$ is represented by a map $A\times U\to B$ for $U\in \Phi$, with two representatives identified when they agree after restriction to some $W\in \Phi$ below both domains. This construction preserves finite products and, when $\mathcal{C}$ is Cartesian closed, exponentials.

If $\mathcal{C}$ has finite categorical limits, every functor $F:\mathcal{C}\to \mathcal{D}$ preserving pullbacks in a category also preserves equalizers, even when it does not preserve a terminal object. Factor it through the slice category $\mathcal{D}/F1$ by sending $A$ to $F(A\to 1)$. This lifted functor preserves pullbacks because the slice forgetful functor reflects them; it also preserves the terminal object, so it preserves finite limits. The slice forgetful functor preserves equalizers, completing the proof.

If a category has pullbacks in a category, each slice category is finitely complete. The identity of the base object is terminal in the slice. Ambient pullbacks in a category inherit a unique common structure map to the base and satisfy the same universal property over that base. A terminal object and pullbacks in a category construct finite products in a category and equalizers, hence all finite limits.

A terminal object and pullbacks in a category give binary products by pulling back the two terminal maps. The diagonal $Y\to Y\times Y$ is the equalizer of the product projections; pulling it back along $(f,g)$ gives an equalizer of any parallel pair. A finite diagram then has a categorical limit as the equalizer of two maps from the product of its object values to the product of its arrow targets. The two maps encode source-image and target projections, so the equalizer enforces every cone equation.

For a nonempty topological space $S$, the constant diagram functor on its open-set presheaf category lies in this five-term adjunction chain. The leftmost functor sends $A$ to the categorical presheaf equal to $A$ at the empty open set and empty elsewhere. The rightmost sends $A$ to the presheaf equal to $A$ at $S$ and a singleton elsewhere. A natural transformation from the former is determined by its empty-open component; a transformation into the latter is determined by its whole-space component. The middle adjunctions follow from restriction maps to the empty open set and from the whole space. Neither endpoint extends: the leftmost functor fails to preserve the terminal object, and the rightmost fails to preserve the initial object.

Every Grothendieck topos has a unique geometric morphism to sets, up to isomorphism. Its inverse image is $\Delta S={\coprod }_{s\in S}1$ and its direct image is $\Gamma A=\mathrm{Hom}(1,A)$. A geometric inverse image out of sets must have this form because it preserves coproducts and the terminal object. A local topos is one for which $\Gamma$ itself is an inverse image.

In a locally small capital regular category, $\Gamma$ is a regular functor and reflects terminal objects. A well-supported object $A$ has a point: otherwise $A\times A$ is well-supported with no points, so its diagonal is invertible by well-pointedness; then $A$ is subterminal and its regular epimorphism to $1$ is invertible, a contradiction. Pulling a regular epimorphism back along a point gives a well-supported object, so the point lifts. This proves preservation of regular epimorphisms, while representability gives limit preservation. If $\Gamma X$ is a singleton, its unique point $1\to X$ is monic and contains all points; $X$ is well-supported, so that point is invertible. Global sections need not reflect arbitrary isomorphisms between objects of proper support.

Use the preservation obstructions to extending an adjoint chain: a functor having a further left adjoint must preserve categorical limits, while one having a further right adjoint must preserve colimits.

At the left end of the presheaf category chain, $\Lambda$ does not preserve the terminal object. The terminal categorical presheaf has singleton value everywhere, whereas $(\Lambda 1)(S)=\varnothing$ because $S\ne \varnothing$. Thus $\Lambda$ cannot itself be a right adjoint. At the right end, $\nabla$ does not preserve the initial object: $(\nabla \varnothing )(\varnothing )=\{\ast \}$ since the empty open set is proper. Thus $\nabla$ cannot itself be a left adjoint.

For the category of small categories chain, take two functors from the terminal category $1$ to the two-object indiscrete category $I\{0,1\}$, selecting its distinct objects. Their equalizer is the empty category. But their images under $C={\pi }_0$ are the same function between singleton sets, whose equalizer is a singleton. Hence $C$ does not preserve equalizers and has no further left adjoint. Finally, $I1⨿I1$ is the two-object discrete category, while $I(1⨿1)$ has arrows between its distinct objects. The canonical comparison is not an isomorphism of categories, so $I$ fails to preserve this coproduct and has no further right adjoint.

Here a universal element means a pair $(A,u)$, with $u\in X(A)$, such that for every $B$ and every $x\in X(B)$ there is a unique morphism $f:B\to A$ satisfying $X(f)(u)=x$. Equivalently,By the Yoneda lemma these component maps constitute a natural transformation $h_A\to X$. If $u$ is universal, every component is bijective, so this is a natural isomorphism and $X$ is a representable presheaf. Conversely a natural isomorphism $h_A\to X$ takes $1_A$ to an element $u$ with exactly this property. In the category of elements of the contravariant functor $X$, $(A,u)$ is a terminal object: every $(B,x)$ has a unique arrow to it.

Given $A,B$, their product in a category is the pullback of the unique arrows $A\to 1\leftarrow B$, where $1$ is a terminal object. Having constructed this product in a category, define the equalizer of $f,g:A\rightrightarrows B$ as the pullbackwhere ${\delta }_B$ is the categorical diagonal. A map $x:X\to A$ factors through this pullback exactly when $fx=gx$: the accompanying arrow to $B$ must equal this common composite. The factorization is unique, proving the equalizer property.

Thus the hypotheses give a terminal object, binary products in a category and all equalizers. Part (i) now gives all finite limits.

Use the explicit construction of small limits from products and equalizers from part (i), here with finite indexing sets. If $F$ preserves finite products in a category and equalizers, then $F(P)$ and $F(Q)$, with their image projections, are the products of the objects $FD(j)$ and of the target objects indexed by arrows of $J$.

The image maps $F(a),F(b)$ have componentsAlso $F(e):F(L)\to F(P)$ is their equalizer. Therefore the same universal property proof makes $F(L)$, with projections $F(p_j)$, a categorical limit of $FD$. Any other chosen limit of $D$ is uniquely isomorphic to this construction, and the functor carries that isomorphism to an isomorphism. Consequently $F$ preserves every finite limit. Preservation of finite products includes the empty product, so the terminal object is also preserved.

First construct every finite product in a category by iterating binary products in a category; the empty product is the terminal object. Let $D:J\to \mathcal{C}$ be a diagram in a category with finitely many objects and arrows. Form the finite products in a categoryDefine $a,b:P\rightrightarrows Q$ by their componentsTake their equalizer $e:L\to P$. Its components $p_j={\pi }_{j}e$ satisfy $D(u)p_i=p_j$, so they form a cone over a diagram.

For any other categorical cone $x_j:X\to D(j)$, the product in a category property supplies a unique $x:X\to P$ with ${\pi }_{j}x=x_j$. The cone equations imply $ax=bx$, so the equalizer property supplies a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. These are exactly the required equations $p_j\bar{x}=x_j$, with uniqueness. Thus $L$ is a categorical limit. For empty $J$, both products are terminal and this construction still gives a terminal limit. All finite limits exist.

Let $D:J\to [\mathcal{C},\mathcal{S}]$ be a small diagram in a category. Since $\mathcal{S}$ is a complete category, choose at each $C\in \mathcal{C}$ a categorical limitFor $f:C\to C^{\prime }$, the arrows $D_j(f)p_{j,C}$ form a cone over a diagram at $C^{\prime }$, because the diagram arrows $D_j\to D_k$ are natural transformations. There is therefore a unique arrow $L(f):L(C)\to L(C^{\prime })$ withThe universal property immediately gives $L(1_C)=1_{L(C)}$ and $L(gf)=L(g)L(f)$: both sides have the same composites with every limit projection. Thus $L$ is a functor, and the displayed equations make the $p_j$ natural transformations.

Given any categorical cone $q_j:X\to D_j$ in the functor category, its components induce unique $q_C:X(C)\to L(C)$. For $f:C\to C^{\prime }$, compare $L(f)q_C$ and $q_{C^{\prime }}X(f)$ after each $p_{j,C^{\prime }}$; naturality of $q_j$ makes their composites equal. Hence $q$ is a natural transformation, and uniqueness is componentwise. This proves that $L$ is a categorical limit in $[\mathcal{C},\mathcal{S}]$.

The functor category is complete, and every evaluation functor preserves limits, since evaluating the constructed limit at $C$ gives exactly the chosen limit in $\mathcal{S}$. This is the pointwise limits in a functor category construction; it also covers the empty diagram and its terminal object.

First prove the exact lifting statement for the forgetful functor $U^T:{\mathcal{C}}^T\to \mathcal{C}$. Take any diagram in a category of algebras for a monad $(A_j,a_j)$ whose underlying diagram has a chosen categorical limit $p_j:L\to A_j$. The arrows $a_{j}T(p_j):TL\to A_j$ form a categorical cone, since every diagram arrow is a morphism of algebras for a monad. Thus the universal property gives a unique arrow $a:TL\to L$ satisfyingThe projections are jointly monic: two arrows into $L$ with the same composites with every $p_j$ are equal. Therefore the algebra identities can be checked after these projections. For the unit,For multiplication,The algebra law for $a_j$ makes the right-hand sides equal. Thus $(L,a)$ is an algebra for a monad, and each $p_j$ is a morphism of algebras for a monad. Its action is unique with this property.

Given an algebra cone $q_j:(B,b)\to (A_j,a_j)$, let $q:B\to L$ be its unique underlying mediating map. Thenso $qb=aT(q)$ by joint monicity. Hence $q$ is an algebra map, proving the lifted cone is limiting. This establishes that the monad algebra forgetful functor creates limits, without any assumption that $T$ preserves limits.

For a monadic adjunction, the comparison satisfies $U=U^{T}K$. If $K$ is an isomorphism of categories over the base, the preceding unique lift transports back literally, so $U$ is a limit-creating functor. If monadic means that $K$ is only an equivalence of categories, the precise conclusion is creation of limits up to isomorphism: the lifted algebra is isomorphic to an object $K(D)$, and its limiting cone transports along that isomorphism. Full faithfulness supplies the arrows and their uniqueness.

This distinction concerns prescribed underlying objects, not existence of limits. Literal creation does not follow from an arbitrary equivalence alone. For example, the inclusion of the one-object category into the two-object indiscrete category, choosing one of its objects, is an equivalence of categories and is monadic under the equivalence convention. Either object of the target is terminal, but the other chosen terminal object has no literal object preimage. Thus the strict reading of the requested assertion needs the strict comparison convention; the equivalence-invariant statement is proved above.

For the Category of sets, take the monoidal tensor product to be the Cartesian product, $A\otimes B=A\times B$, with unit a singleton $1$. On morphisms use $(f\times g)(a,b)=(f(a),g(b))$. Define the associator by $((a,b),c)\mapsto (a,(b,c))$ and the unitors by $(\ast ,a)\mapsto a$ and $(a,\ast )\mapsto a$. These are natural isomorphisms, with inverses obtained by rebracketing or adjoining $\ast$. Both paths in the pentagon send $(((a,b),c),d)$ to $(a,(b,(c,d)))$, and both paths in the triangle send $((a,\ast ),b)$ to $(a,b)$. This proves the axioms elementwise.

There is a second structure: use disjoint union $A⨿B$ and unit $\varnothing$. Define the associator by preserving each element and its summand while changing the nested tags, and define the unitors by removing the empty summand. These are natural and invertible. In the pentagon, either path takes an element in any of the four summands to that same element in the same summand of $A⨿(B⨿(C⨿D))$; the triangle removes an empty intermediate summand on either path. Thus both axioms hold.

These product and coproduct monoidal structures on sets have different units, $1$ and $\varnothing$, which are not isomorphic. They are even inequivalent as monoidal categories: an equivalence of the underlying Category of sets preserves terminal objects, so it cannot take the singleton unit of the product structure to the empty unit of the coproduct structure. Hence the monoidal structure on sets is not unique.

Let $1$ be a terminal object. The pullback in a category of $X\to 1\leftarrow Y$ has precisely the universal property of the binary product in a category $X\times Y$: every pair of morphisms to $X,Y$ is compatible over $1$. Repeated binary products in a category, together with $1$ as the empty product, give all finite products in a category.

For $f,g:X\rightrightarrows Y$, form the pullback in a category of the categorical diagonal ${\Delta }_Y:Y\to Y\times Y$ along $⟨f,g⟩:X\to Y\times Y$. Its projection $e:E\to X$ satisfies $fe=ge$. If $h:T\to X$ satisfies $fh=gh$, the pair $h,fh$ gives a unique morphism into this pullback in a category. Conversely any such morphism must project to that pair. Hence $e$ is the equalizer of $f,g$. The preceding construction now gives all finite limits.

For a functor $D:\mathcal{J}\to \mathcal{C}$ with finite indexing category, form the finite products in a categoryDefine $s,t:P\rightrightarrows Q$ by requiring the coordinate indexed by $u:i\to j$ to be $D(u){\pi }_i$ for $s$ and ${\pi }_j$ for $t$. Let $e:L\to P$ be their equalizer, and put ${\lambda }_j={\pi }_{j}e$. The equality $se=te$ says exactly that $D(u){\lambda }_i={\lambda }_j$ for every morphism $u$, so these morphisms form a categorical cone.

For any other categorical cone $(x_j:X\to D(j))$, the product in a category gives a unique $x:X\to P$ with ${\pi }_{j}x=x_j$. The categorical cone equations give $sx=tx$, so the equalizer gives a unique $\bar{x}:X\to L$ with $e\bar{x}=x$. This is exactly the universal property of a categorical limit. Empty indexing products are terminal objects, so the construction includes the empty diagram in a category. Thus finite products and equalizers give all finite limits.

A localic geometric morphism $p:\mathcal{D}\to \mathcal{F}$ presents $\mathcal{D}$, over $\mathcal{F}$, as sheaves on an internal locale of $\mathcal{F}$. Equivalently, subobjects of inverse images $p^{\ast }X$ form a separating family. A hyperconnected geometric morphism $h$ has a full faithful inverse image $h^{\ast }$ whose essential image is closed under subobjects. In particular, it induces bijectionsFor this definition, subobject-closed means every subobject of $h^{\ast }X$ is isomorphic, over $h^{\ast }X$, to the inverse image of a subobject of $X$.

Here is the construction and proof of the hyperconnected-localic factorization. Given $f:\mathcal{E}\to \mathcal{F}$, its right adjoint gives natural bijectionsLet $L$ be the internal locale whose frame of opens is $f_{\ast }{\Omega }_{\mathcal{E}}$. The frame operations are determined by these bijections: intersections are intersections of represented subobjects; the join of a family is its image under the projection from the parameter-indexed family. For an internal index object $I$, evaluation of the family over $X\times I$ becomes a subobject of $f^{\ast }X\times f^{\ast }I$, and its direct image to $f^{\ast }X$ represents the join. Pullback stability of images makes this natural in $X$, and distributivity follows from the corresponding subobject identity in $\mathcal{E}$. This explicitly constructs the internal frame rather than asserting it from the existence of $f_{\ast }$ alone.

Set $\mathcal{D}={\mathrm{Sh}}_{\mathcal{F}}(L)$, with its localic projection $p$ to $\mathcal{F}$. We construct $h^{\ast }:\mathcal{D}\to \mathcal{E}$. A basic open family is a map $u:X\to f_{\ast }{\Omega }_{\mathcal{E}}$; its represented object in $\mathcal{E}$ is the subobject $A_u\hookrightarrow f^{\ast }X$ corresponding to $u$. Meets and covering joins of open families become intersections and jointly epic families of these subobjects.

An internal sheaf on $L$ is presented by its local sections with their open domains and restrictions. Pull these domains back to the corresponding $A_u$ and identify two sections wherever their restrictions agree. This is an effective equivalence relation in $\mathcal{E}$, so its quotient defines $h^{\ast }$ of the sheaf. More concretely, if $S$ is the object of sections and $d:S\to \mathcal{O}(L)$ gives their domains, start with $A_d\subseteq f^{\ast }S$. For two sections, the join of the opens on which they agree gives an $L$-valued relation on $S\times S$, hence a subobject of $f^{\ast }(S\times S)$. Restricted to $A_d\times A_d$, this is the germ equivalence relation used for the quotient. The restriction and sheaf axioms guarantee reflexivity, symmetry and transitivity. This construction is functorial. Products and equalizers are obtained by meeting domains and equalizing restrictions, so it preserves finite limits. A right adjoint is obtained by taking, over each open family $u$, the parameterized maps $A_u\to Y$ for $Y\in \mathcal{E}$. These maps satisfy the sheaf condition because a covering join gives an effective gluing of the $A_u$; their parameter objects are supplied by the original adjunction $f^{\ast }⊣f_{\ast }$ and exponentials. The map bijection on basic open families, followed by gluing, proves this is right adjoint to $h^{\ast }$. Thus $h$ is a geometric morphism.

We must check hyperconnectedness, not just construct a factorization. A map $A_u\to A_v$ has a graph which is a subobject of $f^{\ast }(X\times Y)$. The represented-subobject bijection turns it into an $L$-valued relation on $X\times Y$. Its totality and single-valuedness say exactly that this relation is a map between the corresponding basic sheaf objects: totality uses the covering join, and single-valuedness uses intersections and the diagonal of $Y$. Thus maps between basic open families are precisely their maps in $\mathcal{E}$. Applying this to the local sections of arbitrary sheaves, compatible maps glue uniquely on both sides. Hence $h^{\ast }$ is full and faithful. Likewise a subobject of $h^{\ast }S$ pulls back to subobjects of its basic $A_u$; each is again represented by an open family, and their compatibility glues to a subsheaf of $S$. The essential image is therefore subobject-closed, proving that $h$ is hyperconnected.

The inverse image $p^{\ast }X$ is the constant family over the whole locale. The construction sends it to $f^{\ast }X$, so $h^{\ast }{p}^{\ast }\cong f^{\ast }$; taking right adjoints gives the required factorizationThis argument uses the usual effective quotient and local-section constructions in a topos; it also explains why the relevant internal opens are relative subobjects of $f^{\ast }X$, not just global subobjects of the terminal object.

For uniqueness, suppose $f=p^{\prime }{h}^{\prime }$ with $h^{\prime }$ hyperconnected and $p^{\prime }$ localic. Then for every $X$,If ${\mathcal{D}}^{\prime }$ is sheaves on an internal locale $L^{\prime }$, the right side is represented by its frame $\mathcal{O}(L^{\prime })$. These identifications preserve intersections, joins and parameter change, so the representing internal frames are isomorphic: $\mathcal{O}(L^{\prime })\cong f_{\ast }{\Omega }_{\mathcal{E}}$. The locale and its sheaf topos are therefore equivalent over $\mathcal{F}$. Under this equivalence the basic open families have the same images in $\mathcal{E}$, and their local-section presentations determine all other images and maps. Thus the two hyperconnected parts agree up to the same equivalence. The factorization is unique up to equivalence.

Finally suppose $f=hi$ with $i:\mathcal{E}\hookrightarrow \mathcal{D}$ a geometric embedding and $h:\mathcal{D}\to \mathcal{F}$ hyperconnected. Let $j$ be the local operator on $\mathcal{D}$ presenting the inclusion. For a subobject $A\hookrightarrow X$ of $\mathcal{F}$, take the $j$-closure of $h^{\ast }A$ in $h^{\ast }X$. Hyperconnectedness makes it the inverse image of a unique subobject of $X$. This defines a closure operation on $\mathcal{F}$; its extensivity, idempotence, intersection preservation and pullback stability descend from the $j$-closure. Let $k$ be its local operator.

Hyperconnectedness also gives $h_{\ast }{\Omega }_{\mathcal{D}}\cong {\Omega }_{\mathcal{F}}$, by the represented-subobject bijections. The direct image $i_{\ast }{\Omega }_{\mathcal{E}}$ is the closed-truth object ${\Omega }_j$, and henceThis follows either by taking the equalizer of $j$ and the identity or by representing closed subobjects in the preceding closure construction. The internal locale with frame ${\Omega }_k$ is the sublocale of the terminal locale determined by $k$: covering joins are exactly closure by $k$. Its sheaves are the $k$-sheaves in $\mathcal{F}$, and its projection is their geometric inclusion. Applying the just-proved factorization to $f$ therefore givesThis supplies the requested reordered factorization.