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A monoid encoding a one-object product and exponential structure: pairing is inverse to the map $z\mapsto (\pi z,{\pi }^{\prime }z)$, and currying is inverse to $y\mapsto \epsilon ⟨y\pi ,{\pi }^{\prime }⟩$. These bijections imply pairing naturality $⟨x,y⟩z=⟨xz,yz⟩$ and currying naturality $(x⟨z\pi ,{\pi }^{\prime }⟩{)}^{\ast }=x^{\ast }z$. The element $e=({\pi }^{\prime }{)}^{\ast }$ then satisfies $ez=e$ for every $z$, hence is idempotent. Splitting $e$ supplies a terminal object and a two-object Cartesian closed category whose endomorphisms at the original object are $M$. For the original object to be nonterminal, $M$ must be nontrivial.

An elementary topos is a category with finite limits, a cartesian-closed structure, and a subobject classifier. These axioms support an internal intuitionistic higher-order logic.

A full subcategory $\mathcal{D}$ of a Cartesian closed category $\mathcal{C}$ is an exponential ideal when $B^A$ belongs to $\mathcal{D}$ for every $B\in \mathcal{D}$ and every $A\in \mathcal{C}$.

For categorical presheaves $P,Q$ on a small category, restrictions in the displayed exponential object act by precomposition with $H_u\times 1_P$. Evaluation sends $(\alpha ,x)$ at $c$ to ${\alpha }_c(1_c,x)$. The currying of $t:R\times P\to Q$ sends $z\in R(c)$ to the natural transformation whose value on $(v:a\to c,x\in P(a))$ is $t_a(R(v)z,x)$. Evaluation and currying are inverse and natural, proving the Cartesian closed category structure.

For a left-exact reflective subcategory of a Cartesian closed category, an ambient exponential $B^A$ is already fixed by the reflector when $A,B$ are fixed. Indeed maps into $B^A$ transpose to maps from $X\times A$ into $B$, and reflection gives $L(X\times A)\cong LX\times A$. Precomposition with the reflection unit is therefore bijective for maps into $B^A$, characterizing a fixed object.

The Scott continuous maps from $D$ to $E$, ordered pointwise. A nonempty directed family has pointwise supremum, and the constant-bottom function is least. Joint evaluation is Scott continuous: for a directed family of pairs $(f_i,x_i)$, directedness makes diagonal values cofinal among all $f_i(x_j)$. Evaluation and currying make the category of pointed complete partial orders and non-strict Scott continuous maps a Cartesian closed category.

A category with finite limits is locally Cartesian closed when every slice category is a Cartesian closed category. Covariant set-valued functor categories on small categories have this property, using the equivalence between their slices and functor categories on categories of elements.

A Cartesian closed category has finite products in a category and, for all $X,Y$, an exponential object $Y^X$ with a natural bijectionEquivalently the functor $-\times X$ has a right adjoint. Its counit of an adjunction is the evaluation map of an exponential object $Y^X\times X\to Y$.

For the presheaf category, define the exponential in a presheaf category byFor $f:D\to C$, its restriction sends $\alpha$ to $\alpha \circ (h_f\times 1_X)$. Composition of these restriction maps follows from composition of the $h_f$, so this is a categorical presheaf. Define evaluation map of an exponential object byFor $f:D\to C$, naturality of $\alpha$ givesThus $e$ is a natural transformation.

Given a natural transformation $\beta :Z\times X\to Y$, define its currying byNaturality of $\beta$ proves that ${\hat{\beta }}_C(z)$ is a natural transformation $h_C\times X\to Y$. For $f:C^{\prime }\to C$, the same formula proves that $\hat{\beta }:Z\to Y^X$ is natural. Evaluating at $g=1_C$ gives $e(\hat{\beta }\times 1_X)=\beta$.

Conversely, if $\gamma :Z\to Y^X$ and $\beta =e(\gamma \times 1_X)$, thenby naturality of $\gamma$. Hence the two constructions are inverse and natural in $Z,Y$. Products are available pointwise, including the terminal singleton presheaf, so the presheaf category is Cartesian closed. The exponential uses natural transformations out of $h_C\times X$; simply exponentiating the individual values generally does not give this object.

An essay arguing against the claim. A good definition can change which mathematical questions are visible. Nevertheless, I would argue that category theory derives its explanatory force from the interaction between definitions and theorems; there is no sound general ordering in which definitions contribute more than theorems.

The strongest case for the contrary view comes from universal properties. A product in a category is characterized by maps into it, rather than by a particular construction of ordered pairs. The same definition applies to sets, groups, and topological spaces. An adjunction goes further: its natural correspondence of arrows puts free constructions, quotient constructions, and several kinds of completion into a common language. Such definitions remove irrelevant choices and expose relationships that would be difficult to recognize from separate constructions.

But a universal-property definition does not supply an object. The definition of a product in a category says what a product would accomplish; an existence theorem says whether the category actually has one. Even uniqueness up to unique isomorphism is a short proof, not an extra clause to be silently supplied by the word “universal”. Given two products, their universal properties produce maps in both directions, and uniqueness forces the two composites to be identities. This elementary theorem makes the definition usable independently of a chosen construction.

The Yoneda lemma illustrates the distinction especially clearly. A representable functor is a definition, but the lemma proves that its natural transformations into any set-valued functor are determined by one element. Its proof evaluates at the identity and reconstructs the whole transformation by functoriality. Consequently the Yoneda embedding is full and faithful: an object's pattern of incoming arrows recovers its morphisms without loss. That is a substantive explanation of why the chosen definition is powerful. Merely calling a functor representable would not establish it.

Similarly, defining a monadic adjunction gives a precise meaning to recovering structured objects as algebras for a monad. It does not decide whether a given forgetful functor is monadic. The precise monadicity theorem translates that difficult classification into a usable test involving adjoint functors, reflected isomorphisms, and specific coequalizers. For compact Hausdorff spaces, the crucial step is proving that a set-split relation is closed. The definition of a monad alone does not contain this topological argument. The theorem connects the algebraic language to an independently meaningful class of spaces.

There is also a danger in praising definitions without hypotheses. A Cartesian closed category has exponential objects, but a category of structured objects need not be cartesian closed merely because underlying sets have function sets. The correct exponential in a functor category generally involves natural transformations out of representable functors, not pointwise function sets. Its existence and universal property require an argument. Theorems also reveal when tempting definitions cannot support the intended construction.

Definitions provide the vocabulary and select the structures worth studying; proofs establish existence, consequences, and limitations. Category theory makes this partnership unusually explicit because its definitions are often expressed through universal properties. Its distinctive achievement is to turn well-chosen definitions into reusable theorems, rather than to make theorems secondary.

An alternative essay arguing for the claim. In category theory, theorems are indispensable, but a carefully chosen definition often performs the greater act of organization. It identifies what many apparently unrelated constructions have in common and makes whole families of proofs possible. In that explanatory sense, definitions can be the subject's most influential contributions.

Consider a universal property. Describing a product in a category as an object representing pairs of incoming arrows replaces several different concrete constructions by one invariant pattern. The corresponding uniqueness proof is important but short; the lasting insight is that the same pattern is the right question in every category. Once the definition is available, it becomes meaningful to ask which functors preserve products, whether the products exist in a new category, and how they interact with other universal constructions.

An adjunction is an even more striking example. The existence of free groups, free modules, and a Stone-Čech compactification can be proved separately. The definition of an adjunction recognizes their common relation to forgetful functors and explains why arrows out of a free object are determined by arrows out of its generators. It also packages the relation as a monad. A large collection of isolated existence results becomes an instance of a shared structure, with reusable consequences about composition and categorical limits.

A valuable definition also has to distinguish structures rather than merely rename them. An abelian category is not just an additive category with categorical kernels and categorical cokernels: normality of monomorphisms and epimorphisms matters. The additive example with truncated primary torsion shows why those clauses belong in the definition. The definition therefore preserves exactly the structure on which theorems about images and exact sequences depend.

This view does not make proofs dispensable. The Yoneda lemma justifies the use of representable functors, and the precise monadicity theorem tests when a proposed algebraic description is valid. Rather, it argues that the best categorical definitions have unusually broad explanatory reach: they select the invariant structures to which those proofs apply. If “matter more” means providing the reusable framework in which many theorems become instances of one idea, category-theoretic definitions have a strong claim to that role.

For a locally small category $\mathcal{C}$, an object $A$, and a functor $F:\mathcal{C}\to \mathbf{Set}$, the covariant form of the Yoneda lemma is the natural bijectionFor $x\in F(A)$ define ${\alpha }_B^x(f)=F(f)(x)$, where $f:A\to B$. For $u:B\to D$, functoriality gives $F(u){\alpha }_B^x(f)=F(uf)(x)={\alpha }_D^x(uf)$, so this is a natural transformation. Conversely, naturality of $\alpha$ at $f:A\to B$ gives ${\alpha }_B(f)=F(f){\alpha }_A(1_A)$. The two constructions are inverse. They are natural in $F$, because a natural transformation $F\Rightarrow G$ commutes with evaluation. They are also natural in $A$: an arrow $v:A^{\prime }\to A$ induces $\mathcal{C}(A,-)\to \mathcal{C}(A^{\prime },-)$ by precomposition, and the resulting map $\mathrm{Nat}(\mathcal{C}(A^{\prime },-),F)\to \mathrm{Nat}(\mathcal{C}(A,-),F)$ corresponds to $F(v):F(A^{\prime })\to F(A)$. Applying the result to the opposite category gives $\mathrm{Nat}(\mathcal{C}(-,A),H)\cong H(A)$ for $H:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$.

Now let $\mathcal{C}$ be a small category, and let $F,G:\mathcal{C}\to \mathbf{Set}$. Finite products in a category in $[\mathcal{C},\mathbf{Set}]$ exist pointwise; the terminal functor is the constant singleton functor. Define the exponential of covariant set-valued functors bySmallness ensures that these natural transformations form a set. For $u:C\to D$, define $(G^F)(u)(\alpha )$ at $E$ byPrecomposition respects identities and composition, so this defines a functor. Its evaluation map of an exponential object isTo check naturality, for $u:C\to D$ both sides give ${\alpha }_D(u,F(u)x)$: one side uses naturality of $\alpha$, and the other uses the defining action of $G^F$.

Here is the full universal property. Given $\theta :H\times F\Rightarrow G$, define $\hat{\theta }:H\Rightarrow G^F$ byFor $w:D\to E$, naturality of $\theta$ makes this commute with $G(w)$, so ${\hat{\theta }}_C(h)$ is a natural transformation. Replacing $h$ by $H(u)h$ is the same as precomposing $v$ by $u$, proving naturality in $C$. Conversely, $\beta :H\Rightarrow G^F$ gives ${\theta }_C(h,x)={\beta }_C(h{)}_C(1_C,x)$. Starting with $\theta$ recovers it immediately. Starting with $\beta$, its naturality at $v:C\to D$ yieldsso it too is recovered. The formulas are natural in $H$ and $G$, giving the required adjunction $(-)\times F⊣(-{)}^F$. Therefore the covariant functor category is cartesian closed. This proof supplies the actual exponential and does not need a colimit presentation of its arguments.

Finally suppose $\mathcal{C}$ itself is a small category which is a Cartesian closed category. In its presheaf category, the same construction, applied to ${\mathcal{C}}^{\mathrm{op}}$, givesThe first isomorphism uses the product's universal property: a map into $C\times A$ is precisely a pair of maps into $C$ and $A$. The next is the Yoneda lemma, and the last is the exponential adjunction in $\mathcal{C}$. Each step is natural in $C$, $A$, and $B$. Under these identifications, evaluation corresponds to $Y$ of the evaluation $B^A\times A\to B$. Hence the Yoneda embedding preserves exponentials, with the natural identification $\boxed{Y(B^A)\cong (YB{)}^{YA}}$.

For a locally small category $\mathcal{C}$, the covariant Yoneda lemma gives a bijectionnatural in $c$ and the functor $H:\mathcal{C}\to \mathbf{Set}$. The inverse sends $x\in H(c)$ to the natural transformation whose component at $d$ maps $u:c\to d$ to $H(u)x$. Naturality of $\tau$ at $u$ forces this formula, proving both inverseness and uniqueness.

Write $\mathcal{E}=\int F$ for the category of elements of $F$. Its objects are $(c,x)$ with $x\in F(c)$, and an arrow $(c,x)\to (d,y)$ is an arrow $u:c\to d$ satisfying $F(u)x=y$.

Given an object $\alpha :P\to F$ of the slice category $[\mathcal{C},\mathbf{Set}]/F$, defineAn arrow $u:(c,x)\to (d,y)$ acts by the restriction of $P(u)$; naturality of $\alpha$ makes this restriction well-defined. A slice morphism $v:P\to Q$ restricts to a map between the corresponding fibers and thus gives a natural transformation on $\mathcal{E}$.

Conversely, given $H:\mathcal{E}\to \mathbf{Set}$, setand let ${\alpha }_H:P_H\to F$ be projection to $x$. The functor laws and naturality follow from those for $F$ and $H$. On morphisms use the maps on each summand. Taking a fiber of this disjoint union recovers $H(c,x)$ canonically. Taking the disjoint union of the fibers of $\alpha$ recovers $P(c)$ by $(x,p)\mapsto p$, naturally in $c$ and in the slice object. These are inverse natural isomorphisms, provingThis is the slice of a set-valued functor category construction.

Now assume $\mathcal{C}$ is a small category, so $\mathcal{E}$ is small. Define a diagram in a categoryFor $u:(c,x)\to (d,y)$ in $\mathcal{E}$, its reverse arrow acts by precomposition $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. There is a cocone to $F$ with component $g\mapsto F(g)x$. To verify its colimit property directly, use pointwise colimits of sets. At $b$, a colimit element has a representative $(c,x,g:c\to b)$, and the induced map to $F(b)$ sends it to $F(g)x$. This map is surjective, since $z\in F(b)$ is represented by $(b,z,1_b)$. Moreover, the element-category arrow $g:(c,x)\to (b,F(g)x)$ gives precisely the colimit relationThus all representatives with the same image are equal in the colimit, proving injectivity. The resulting bijections are natural in $b$. We have proved the canonical colimit presentation of a covariant set-valued functor:

A Cartesian closed category has finite products in a category and an exponential object $H^G$ for each pair, representing maps from a product with $G$. In $[\mathcal{C},\mathbf{Set}]$, the terminal functor and binary products are computed pointwise. Define the exponential of covariant set-valued functors bySmallness makes this a set. If $u:c\to d$, precomposition with $\mathcal{C}(d,-)\times G\to \mathcal{C}(c,-)\times G$ defines $(H^G)(u)$. Functoriality is immediate from associativity of composition. Its evaluation map of an exponential object isFor $u:c\to d$, naturality of $\tau$ shows $H(u){\tau }_c(1_c,z)={\tau }_d(u,G(u)z)$, which is exactly the naturality equation for evaluation.

Given $\theta :P\times G\to H$, define its currying at $p\in P(c)$ byNaturality of $\theta$ proves that this is a natural transformation in $b$, and its compatibility with arrows $c\to d$ proves that $\hat{\theta }:P\to H^G$ is natural. Evaluation recovers $\theta$ by setting $f=1_c$. Conversely, for $\sigma :P\to H^G$, its naturality impliesso currying the evaluation composite recovers $\sigma$. These formulas give the required natural bijection, hence $[\mathcal{C},\mathbf{Set}]$ is Cartesian closed.

Finally, for every $F$ the small category $\mathcal{E}=\int F$ has a Cartesian closed functor category by the same construction. The slice equivalence above transports its terminal object, products and exponentials to $[\mathcal{C},\mathbf{Set}]/F$. The original functor category also has pointwise finite limits. Therefore it is locally Cartesian closed. Every construction here is explicit; no adjoint functor theorem is used.

Choose mutually inverse maps $i:D^D\to D$ and $j:D\to D^D$. This is an extensional reflexive object in the Cartesian closed category. Define its application morphismFor endomorphisms $f,g:D\to D$, application in the underlying set $\mathrm{End}(D)$ is $f\cdot g=\mathrm{app}\circ ⟨f,g⟩$.

It is useful to define term interpretation uniformly over any context object $A$. An environment assigns each free variable $x$ a morphism $\rho (x):A\to D$. Variables are interpreted by their assigned maps, and application is interpreted by $\mathrm{app}$ and pairing. For lambda abstraction, lift the old environment along ${\pi }_A:A\times D\to A$, interpret the new variable by ${\pi }_D$, and let $h:A\times D\to D$ interpret the body. Then setTaking $A=D$ gives interpretations in the requested set of endomorphisms. Other context objects are needed in the recursive definition to handle bound variables correctly.

The beta reduction rule follows from $ji={\mathrm{id}}_{D^D}$ and the evaluation identity for currying. Applying the interpreted abstraction to a map $g:A\to D$ givesInduction on terms identifies this with interpretation of the capture-avoiding substitution of the argument for $x$.

For eta conversion, suppose $x$ is not free in $M$ and write $f:A\to D$ for the interpretation of $M$. The body $Mx$ on $A\times D$ is the uncurried form of $jf$. HenceThus beta-eta-equivalent terms have equal interpretations in $\mathrm{End}(D)$. The full isomorphism, rather than just a retract of the function object, provides the inverse law used in the eta step.

A Cartesian closed category has finite categorical products and, for every $P,Q$, an exponential object $Q^P$ with an evaluation morphism $\mathrm{ev}:Q^P\times P\to Q$ such that composition with evaluation gives natural bijectionsFinite categorical products in the presheaf category are pointwise; in particular its terminal object is the constant singleton categorical presheaf. For categorical presheaves $P,Q$ defineFor $u:d\to c$, restriction sends $\alpha$ to $\alpha \circ (H_u\times 1_P)$. Composition and identity laws follow from those of the Yoneda embedding, so this is a categorical presheaf.

Define evaluation at $c$ by ${\mathrm{ev}}_c(\alpha ,x)={\alpha }_c(1_c,x)$. For $u:d\to c$, naturality of $\alpha$ gives $Q(u){\alpha }_c(1_c,x)={\alpha }_d(u,P(u)x)$, which is the evaluation of the restricted pair. Hence evaluation is a natural transformation.

Given $t:R\times P\to Q$, its currying sends $z\in R(c)$ to the transformation whose component at $a$ isNaturality in $a$ follows from that of $t$, and naturality in $c$ follows by composing $v$ with the relevant arrow. Evaluating $\hat{t}$ at $(1_c,x)$ recovers $t_c(z,x)$. Conversely, if $s:R\to Q^P$, naturality of $s$ along $v:a\to c$ says that evaluating $s_a(R(v)z)$ at $(1_a,x)$ gives $(s_c(z){)}_a(v,x)$. Thus currying the evaluation of $s$ recovers $s$. These constructions are inverse and natural, proving the universal property. This is the exponential in a presheaf category, and proves the presheaf category is Cartesian closed.

Write a coalgebra for a comonad as $(A,a)$ with $a:A\to GA$, ${ϵ}_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. Since the Cartesian comonad $G$ preserves finite limits, the forgetful functor $U:{\mathcal{E}}_G\to \mathcal{E}$ creates them: for an underlying limit $K$ of coalgebras, the maps $K\to A_i\to G{A}_i$ induce a unique $K\to GK$ using $GK\cong {\lim }_{i}G{A}_i$. Its counit and coassociativity equations follow after every limiting projection, which jointly detect equality. The same argument makes the limiting cone universal among coalgebra cones. Thus ${\mathcal{E}}_G$ is a Cartesian category; in particular, the product structure on $A\times B$ is $a\times b$ followed by the comparison $GA\times GB\cong G(A\times B)$.

An exponential object needs more than the underlying ambient $B^A$. The cofree coalgebra functor is $R(D)=(GD,{\delta }_D)$, with adjunctionThe inverse sends $\tilde{h}$ to ${ϵ}_D\tilde{h}$; the comonad laws verify that these are inverse and that $\tilde{h}$ is a coalgebra map.

For $(A,a)$ and $(B,b)$, put $D=B^A$ and $W=(GB{)}^A$ in the ambient Cartesian closed category. Define $p,q:GD\to W$ byand by transposing the compositeto obtain $q$. Their cofree-adjoint transposes are coalgebra mapsTake their coalgebra equalizer $e:H\hookrightarrow R(D)$, which exists by the finite-limit construction.

To verify its universal property, let $h:C\to D$ transpose to $t:C\times A\to B$. Its coalgebra transpose $\tilde{h}=Ghc$ factors through $H$ exactly when $p^{♯}\tilde{h}=q^{♯}\tilde{h}$. By the cofree adjunction, this equality is equivalent to $p\tilde{h}=q\tilde{h}$. The first side is $b^{A}h$, because ${ϵ}_{D}Ghc=h$; transposing the second side gives $Gt$ composed with the product coalgebra structure on $C\times A$. The equality therefore says exactlywith the canonical product comparison understood: precisely the condition that $t$ is a coalgebra map. Thusnaturally in the coalgebra $C$. Its evaluation is ambient evaluation after ${ϵ}_{D}e\times 1_A$, and the preceding condition makes it a coalgebra map. This equalizer is the coalgebra exponential, so the category of coalgebras is cartesian closed. This Cartesian closure for a finite-limit-preserving comonad uses no subobject classifier on $\mathcal{E}$.

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

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.

The Yoneda lemma states that for $A\in \mathcal{C}$ and $P:\mathcal{C}\to \mathbf{Set}$ there is a natural bijectionIf $P\twoheadrightarrow Q$ is an epimorphism in the functor category, it is pointwise surjective, so $P(A)\to Q(A)$ is surjective. Yoneda identifies this map withThus every representable functor is a projective object in a category.

The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family into a locally small category has a right adjoint functor. For small $\mathcal{C}$, the category $[\mathcal{C},\mathbf{Set}]$ is locally small and has colimits pointwise. Quotients of $P$ are represented by compatible equivalence relations on the sets $P(C)$, so they form a set; hence the category is well-copowered. The set of representables $\{\mathcal{C}(C,-):C\in \mathcal{C}\}$ generates it by the Yoneda lemma. The theorem therefore gives a right adjoint to every small-colimit-preserving functorIn particular, product with a fixed functor $P$ is computed pointwise, and $-\times P(C)$ preserves colimits in the Category of sets. Hence $-\times P$ preserves all small colimits and has a right adjoint $(-{)}^P$. Thus $[\mathcal{C},\mathbf{Set}]$ is a Cartesian closed category.

Now work in $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and write $yA=\mathcal{C}(-,A)$. If $\mathcal{C}$ has binary products, thenThus exponentiation by $yA$ is precomposition with $-\times A$. Precomposition between functor categories has a right adjoint given by Right Kan extension, so $yA$ is a tiny object.

Conversely, suppose $\mathcal{C}$ has a terminal object $1$ and $F$ is tiny. The representable $y1$ is the terminal presheaf, and the exponential adjunction plus Yoneda givesSince $F$ is tiny, $(-{)}^F$ is a left adjoint and preserves all colimits; evaluation at $1$ also preserves pointwise colimits. Therefore the hom functor $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}](F,-)$ preserves coproducts and epimorphisms. Preservation of epimorphisms makes $F$ projective, while preservation of coproducts makes it indecomposable.

A categorical limit of $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone: it consists of an object $L$ and compatible maps $L\to D(j)$ through which every other cone factors uniquely. For a finite diagram, take the product $P={\prod }_{j}D(j)$ and the product $Q={\prod }_{u:i\to j}D(j)$. There are two maps $P\rightrightarrows Q$: on the $u:i\to j$ coordinate one uses respectively $D(u)$ after projection to $D(i)$ and direct projection to $D(j)$. Their equalizer is exactly the compatible-cone object. Hence finite products and equalizers construct every finite limit.

In the category of metric spaces and non-expansive maps, give $X\times Y$ the maximum metricThe projections are non-expansive, and a pair of non-expansive maps into $X$ and $Y$ induces a non-expansive map into this product, proving the universal property.

Let two maps $1\rightrightarrows X$ select $a,b\in X$. On the set quotient identifying $a$ and $b$, define the quotient metric by shortest paths that may jump from $a$ to $b$ at zero cost. Explicitly,This is the largest metric making the quotient map non-expansive. A map $X\to Z$ equalizing $a$ and $b$ factors through the set quotient and remains non-expansive by the path formula, so this is the coequalizer. Its underlying set is the set-theoretic coequalizer.

For $X=\{0,1,5,6\}$ with $0\sim 6$, write $\ast =\{0,6\}$. The quotient hasThus in $Q\times Y$, with $Y=\{0,3\}$,After first taking $X\times Y$, the product of the parallel pair identifies $(0,y)$ with $(6,y)$ separately for $y=0,3$. In its quotient metric, every path from $(1,0)$ to $(5,3)$ has length at least $4$, and length $4$ is attained either directly or via one of those identifications. The canonical bijection from this coequalizer to $Q\times Y$ is therefore not an isometry. Hence $-\times Y$ does not preserve this coequalizer. In a Cartesian closed category, $-\times Y$ is a left adjoint and preserves all colimits, so $\mathbf{Met}$ is not cartesian closed.

For a reflective subcategory $\mathcal{D}$ of a Cartesian closed category $\mathcal{C}$, with reflector $L$, the subcategory $\mathcal{D}$ is an exponential ideal if and only if the canonical comparisonis an isomorphism for every $A,B$. The proof repeatedly curries a map into an object of $\mathcal{D}$ and factors it through the unit of the reflection.

An object $A$ of a Cartesian closed category is tiny when exponentiation $(-{)}^A$ itself has a right adjoint.

The extra adjunction defining a tiny object should not be confused with the filtered-colimit condition for a compact object (mathematics).