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Choose one regular functor witnessing the regular-logic separation of a proper subobject for each proper subobject represented by a monomorphism of the small regular category. Their product is regular and reflects invertibility of monomorphisms. If an arbitrary morphism becomes invertible, its image inclusion becomes invertible and so does the diagonal of its kernel pair. The original morphism is both a regular epimorphism and a monomorphism, hence invertible. The target is a power of sets as a capital regular category. This proves an isomorphism-reflecting regular representation without asserting full faithfulness.

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.

Under the hypotheses of the limit form of the special adjoint functor theorem, intersect all subobjects of $C$ supporting $x:B\to GC$. Limit preservation gives a smallest supporting $(C_0,x_0)$. If $G(u)x_0=G(v)x_0$ for maps $u,v:C_0\to Q_i$, their equalizer supports $x_0$, so minimality makes it invertible and $u=v$. Thus the indicated map of hom-sets is injective. The evaluation embedding into cogenerator products embeds $C_0$ in a product of cogenerators indexed by the realized subsets of the fixed sets $\mathcal{D}(B,G{Q}_i)$. There are only a set of such products, a set of their subobjects, and a set of maps from $B$ into their images under $G$. These data give a weakly initial set in $(B\downarrow G)$. Using realized subsets avoids assuming maps into every cogenerator exist.

A functor between coherent categories is coherent when it preserves finite limits, image factorizations and finite unions of subobjects, including the empty union. It consequently preserves the interpretation of every coherent formula. A coherent functor from a coherent syntactic category to a topos is equivalent to a model of the corresponding coherent theory; natural transformations correspond to ordinary model homomorphisms.

An object is decidable when its diagonal is a complemented subobject of its square. Equivalently its internal equality is decidable. Pullback gives closure under subobjects; coordinatewise equality gives closure under finite products. For an existing coproduct, unequal summands together with the complements of the diagonal in each equal summand form the diagonal complement.

The pullback of the truth point along a local operator $j$ is the subobject $J\hookrightarrow \Omega$. A monomorphism with characteristic map $\chi$ is j-dense exactly when $\chi$ factors through $J$. Its internal order is inherited from the Heyting algebra of truth values. Having a least global element is an internal condition over all parameter objects, not just a condition on global truth values.

A monomorphism $m$ is extremal if every factorization $m=he$ with $e$ an epimorphism forces $e$ to be an isomorphism. A strong monomorphism has this property by its lifting property. In a complete category that is a well-powered category, intersecting all strong subobjects containing a given morphism yields an epimorphism–strong-monomorphism factorization; applying it to an extremal monomorphism proves that the extremal monomorphism is strong. The factorization definition and lifting definition should be kept distinct until the needed ambient hypotheses are supplied.

In a category with pullback in a category constructions and image factorizations, direct image ${\exists }_f(A^{\prime })=\mathrm{im}(A^{\prime }\hookrightarrow A\stackrel{f}{\to }B)$ is a left adjoint to inverse image on subobjects. Frobenius reciprocity is ${\exists }_f(A^{\prime }\cap f^{\ast }{B}^{\prime })={\exists }_f(A^{\prime })\cap B^{\prime }$. It holds for all such subobjects exactly when strong epimorphisms are stable under pullback along monomorphisms. For sufficiency pull the strong part of the image factorization back along the mono into its intersection with $B^{\prime }$; for necessity take $A^{\prime }=A$ and $f$ strong, whose image is the whole codomain.

A set of sequents between geometric formulas over a many-sorted signature. Its geometric syntactic category and geometric syntactic topology yield its classifying topos. A model satisfies a sequent when the interpreted antecedent subobject is contained in the consequent in the indicated context.

An abelian category with enough projectives is hereditary when every subobject of a projective object is projective. For $0\to K\to P\to A\to 0$ with $P$ projective, dimension shifting gives $L_{2}F(A)\cong L_{1}F(K)$. Hence hereditaryness is equivalent to vanishing of $L_{2}F$ for every additive right exact functor into any abelian category, by projectivity detected by all first left derived functors. It is also equivalent to left exactness of every $L_{1}F$: the derived long exact sequence gives injectivity of $L_{1}F(K)\to L_{1}F(P)$ when $L_{2}F$ vanishes, and applying left exactness to the displayed sequence proves the converse.

Meets of subsheaves are pointwise intersections. Joins are J-closures of pointwise unions: a section lies in the join when its restrictions locally belong to some member, with the member allowed to vary. Implication consists of sections every restriction of which belongs to $D$ whenever it belongs to $A$; this gives $C\le (A\Rightarrow D)$ exactly when $C\cap A\le D$. Negation is implication into the initial subobject. Empty covers mean that the initial sheaf need not be pointwise empty.

A geometric morphism $h$ is hyperconnected when its inverse image functor of a geometric morphism is full and faithful and its essential image is closed under subobjects. It then identifies all subobjects of $h^{\ast }X$ with subobjects of $X$, and its image is closed under quotients as well, by descending the kernel-pair relation.

An image of $f:A\to B$ is the least subobject $m:I\hookrightarrow B$ through which $f$ factors. In a category with pullback in a category constructions this gives a strong epimorphism–monomorphism factorization $f=me$: pull back any mono in a lifting square for $e$ to $I$; minimality of the image forces the resulting mono to be invertible, producing a diagonal. Conversely, the lifting property of the strong part proves minimality. No stability under pullback is asserted; that is an additional property in a regular category.

The subobject classifier supplies intuitionistic truth values. Subobjects interpret predicates, finite limits interpret finite conjunctions and equality, and suitable image and adjoint constructions interpret quantifiers. One must distinguish internal intuitionistic arguments from classical reasoning in the external category of sets.

In a complete category, the intersection of a set-indexed family of subobjects of $C$ is their wide pullback in a category over $C$. The resulting morphism into $C$ is a monomorphism, and it factors through every member. In a well-powered category, even the intersection of all subobjects satisfying a specified property is small.

For a limit-preserving functor $U$ and an arrow $d:D\to UC$, a supported subobject is $m:M\hookrightarrow C$ through which $d$ factors after applying $U$. In a complete well-powered domain, their intersection remains supported. At the resulting pair $(C_0,d_0)$, every supported subobject of $C_0$ is invertible. This minimality makes $q\mapsto U(q)d_0$ injective on morphisms to each cogenerator.

Only $L\Rightarrow A$ remains. Assume $U$ preserves small categorical limits, and choose a small cogenerating family $(Q_i{)}_{i\in I}$ in $\mathcal{C}$. Fix a set $X$. The comma category $(X\downarrow U)$ is a complete category: underlying limits in $\mathcal{C}$, together with limit preservation by $U$, lift compatible maps from $X$. It is locally small, since its arrows form subsets of the hom-sets in $\mathcal{C}$.

We construct a weakly initial set in this comma category. Given $f:X\to U(C)$, intersect all subobjects $m:B\hookrightarrow C$ through which $f$ lifts under $U$. There are only a set of subobjects because $\mathcal{C}$ is well-powered; the family includes $1_C$. Their intersection exists by completeness. Also $U$ preserves monomorphisms, since a map is monic precisely when its self-pullback diagonal is invertible, and $U$ preserves such pullbacks.

Thus all lifts of $f$ are unique and compatible, and preservation of the intersection givesNo proper subobject of $C_0$ supports a lift of $f_0$: its composite into $C$ would belong to the intersected family, forcing it to contain $C_0$. This is a minimal supported subobject.

For each $i$, the mapis injective. Indeed, equal images mean $f_0$ lifts to the equalizer of the two maps, because $U$ preserves that equalizer. Minimality forces the equalizer to be invertible, so the maps agree.

The cogenerator property makes the evaluation mapa monomorphism: any two arrows into $C_0$ that agree after all these projections are equal. The indexing pairs $(i,h)$ inject into the fixed setTheir image is some subset $J\subseteq B_X$. Consequently $C_0$ is isomorphic to a subobject of a product $P_J$ of the $Q_i$ indexed by $J$. There are only a set of such subsets $J$, only a set of subobjects of each $P_J$, and only a set of maps $X\to U(B)$ for each representative subobject $B$.

Collect all these pairs $(B,x:X\to U(B))$. Every $(C,f)$ receives a comma arrow from one of them, via the isomorphic copy of $(C_0,f_0)$ and $m_0$. Hence they form a weakly initial set. The initial-object lemma for complete categories with a weakly initial set, proved explicitly in Question 8, gives an initial object of $(X\downarrow U)$. Its map from $X$ is a universal arrow from an object to a functor. Choosing these universal arrows for all $X$ defines a left adjoint to $U$.

ThusThe products indexed by realized subsets $J$ are important: one cannot assume that $C_0$ has a map to every cogenerator and simply add arbitrary missing coordinates.