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The representing object in the functor category $[J,\mathcal{C}]$ is the constant diagram in a category ${\Delta }_{J}C$. If $p_j:{\lim }_j{E}_j\to E_j$ is the chosen categorical limit cone, the natural bijection isA natural transformation from the constant diagram in a category is precisely a categorical cone with vertex $C$, and its inverse is the unique mediating morphism from the universal property of the categorical limit. For $\alpha :E\Rightarrow E^{\prime }$, the defining equations for $\lim \alpha$ show that postcomposition by $\lim \alpha$ corresponds to postcomposition by $\alpha$ on the right. This establishes naturality in $E$ and proves representability by the constant diagram.

Suppose $F\cong {\mathrm{colim}}_{j\in J}{R}_j$, where $J$ is a filtered category and every $R_j$ is a representable functor. Every covariant representable functor $\mathcal{C}(A,-)$ preserves all existing categorical limits: maps from $A$ into a limiting object are exactly compatible families of maps into its diagram.

For a finite diagram in a category $K:I\to \mathcal{C}$, use pointwise colimits in the functor category, preservation by each $R_j$, and the permitted result that filtered colimits commute with finite limits in sets to obtainThe isomorphism is the canonical comparison, since every step respects the limiting projections. This also covers the empty finite diagram and hence the terminal object. Thus $(iv)\Rightarrow (i)$, completing the cycle:

Write $h_A=\mathcal{C}(A,-)$. The covariant Yoneda lemma gives the natural bijectionwhose inverse sends $x\in F(A)$ to the natural transformation with component $u:A\to B\mapsto F(u)(x)$. This also identifies the two variables' naturality.

To prove that $h_A$ is a projective object in a category, let $p:G\twoheadrightarrow F$ be an epimorphism in the functor category and $\alpha :h_A\to F$. By pointwise epimorphism in a functor category, lift ${\alpha }_A(1_A)$ through the surjective function $p_A$. The Yoneda lemma turns this lift into $\beta :h_A\to G$ with $p\beta =\alpha$.

For a small category, the set of pairs $(A,x)$ with $x\in F(A)$ indexes the canonical mapIt is natural, and every $y\in F(B)$ is the image of $(B,y,1_B)$. Thus it is an epimorphism, establishing the projective cover of a set-valued functor by representables construction. Here cover means a projective presentation; no minimality is asserted.

For the equivalence's first step, if $f:B\to D$ is a monomorphism, then $h_A(f)$, the map $u\mapsto fu$, is injective for every $A$. Hence every representable functor is a monofunctor. Conversely, injectivity of every $h_A(f)$ is exactly left cancellation against every $u,v:A\to B$, so it implies that $f$ is a monomorphism. Therefore

Let $P$ be a tiny object of $[\mathcal{C},\mathbf{Set}]$, and let $0$ be the initial object of $\mathcal{C}$. The representable functor $h_0=\mathcal{C}(0,-)$ is the terminal functor, since each of its values is a singleton. The exponential adjunction and the Yoneda lemma giveSince $P$ is tiny, $E_P:F\mapsto F^P$ is itself a left adjoint, so preserves all colimits. Evaluation at $0$ also preserves colimits, because they are pointwise in a set-valued functor category. Therefore $\mathrm{Nat}(P,-)$ preserves the coproduct construction.

It also preserves epimorphisms. Indeed, every such epimorphism is a pointwise epimorphism in a functor category, hence the coequalizer of its kernel pair, as can be checked pointwise in sets. A left adjoint preserves that coequalizer and therefore sends it to a regular epimorphism, which is an epimorphism; evaluation at zero preserves surjectivity. Thus $P$ is an irreducible projective in a set-valued functor category.

A Cauchy-complete category is one in which every idempotent morphism splits. By the characterization allowed in the question, irreducible projectives are representable functors for such an indexing category. Applying it to $P$ proves the tiny covariant functor on a Cauchy-complete category with an initial object is representable conclusion:

A monad on $\mathcal{C}$ is an endofunctor $T$ with natural transformationssatisfying $\mu T\mu =\mu \mu T$ and $\mu T\eta =1_T=\mu \eta T$. Its Eilenberg-Moore category ${\mathcal{C}}^T$ has algebras for a monad $(A,a:TA\to A)$ satisfyingand morphisms of algebras for a monad $f:(A,a)\to (B,b)$ satisfying $fa=bT(f)$.

The Kleisli category ${\mathcal{C}}_T$ has the objects of $\mathcal{C}$ andIts identity is ${\eta }_A$, while the composite of $f:A\to TB$ and $g:B\to TC$ is ${\mu }_{C}T(g)f$. The free functor into a Kleisli category $F_T:\mathcal{C}\to {\mathcal{C}}_T$ is the identity on objects and sends $f:A\to B$ to ${\eta }_{B}f$.

On the functor category $[\mathcal{D},\mathcal{C}]$, postcomposition gives the pointwise monad on a functor categoryThe monad laws hold componentwise. A $T_{\ast }$-algebra is a functor $F:\mathcal{D}\to \mathcal{C}$ with a natural transformation $a:TF\to F$ whose components are $T$-algebras. Naturality says precisely that every $F(u)$ is an algebra morphism. Hence sending $(F,a)$ to the lifted functor $\mathcal{D}\to {\mathcal{C}}^T$ gives an isomorphism, and in particular an equivalence,

On $[\mathcal{C},\mathcal{D}]$, precomposition gives the precomposition monad on a functor categoryA $T^{\ast }$-algebra is a natural transformation $a:GT\to G$ satisfying $aG\eta =1_G$ and $a(aT)=aG\mu$. From it define $\bar{G}:{\mathcal{C}}_T\to \mathcal{D}$ byThe two algebra laws say exactly that $\bar{G}$ preserves identities and Kleisli composition, and $\bar{G}{F}_T=G$. Conversely, a factorization $G=\bar{G}{F}_T$ giveswhere $1_{TA}$ represents a Kleisli arrow $TA\to A$. These constructions are inverse on objects and morphisms, so

For $A\in \mathcal{C}$ and $F:\mathcal{C}\to \mathbf{Set}$, the Yoneda lemma gives a natural bijectionAssume $\mathcal{C}$ is small. For every pair $(A,x)$ with $x\in F(A)$, let ${\alpha }^{A,x}:\mathcal{C}(A,-)\to F$ be the corresponding natural transformation. Their copairing isAt an object $B$, the element $x\in F(B)$ is the image of $1_B$ in the summand indexed by $(B,x)$. The map is therefore pointwise surjective and hence an epimorphism in the functor category.

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 category is well-powered when the isomorphism classes of monomorphisms into each object form a set. For $P\in [\mathcal{C},\mathbf{Set}]$, every subobject is represented by a subfunctor $S$ with $S(A)\subseteq P(A)$. Since $\mathcal{C}$ is small, all choices lie in the set ${\prod }_A\mathcal{P}(P(A))$, and naturality merely cuts out a subset. Thus the functor category is well-powered. Quotients are similarly represented by compatible equivalence relations on the sets $P(A)$, so they form a subset of ${\prod }_A\mathcal{P}(P(A{)}^2)$; hence it is well-copowered.

A cocone under the identity diagram consists of maps $c_A:A\to L$ satisfying $c_{B}f=c_A$ for every $f:A\to B$. A terminal object supplies the unique such cocone and has the required universal property. Conversely, if $(L,c_A)$ is a colimit of the identity, both $1_L$ and $c_L$ mediate its cocone to itself, so uniqueness gives $c_L=1_L$. For any $f:A\to L$, cocone compatibility gives $f=c_{L}f=c_A$; thus there is exactly one arrow $A\to L$, and $L$ is terminal.

For $A\in \mathcal{C}$, let $\Omega (A)$ be the set of isomorphism classes of quotients of the representable $\mathcal{C}(A,-)$. This is a set by well-copoweredness. A map $u:A\to B$ sends a quotient of $\mathcal{C}(A,-)$ to the image quotient of the composite $\mathcal{C}(B,-)\to \mathcal{C}(A,-)$, making $\Omega$ a functor. For any functor $P$ and $x\in P(A)$, Yoneda gives $\mathcal{C}(A,-)\to P$; factor it as an epimorphism followed by a monomorphism and send $x$ to the resulting quotient class. These maps $P\to \Omega$ agree along every monomorphism. Conversely, a cone with apex $Q$ assigns to a quotient $q:\mathcal{C}(A,-)\twoheadrightarrow R$ the element obtained by applying its leg at $R$ to $q_A(1_A)$. Yoneda and epi-mono factorization show that this is well-defined and is the unique map $\Omega \to Q$. Therefore $\Omega$ is a local state classifier.

An object of $[\mathbb{Z},{\mathbf{Set}}_f]$ is a finite set with a permutation. For each $n>0$, let $A_n=\mathbb{Z}/n$ with trivial action and map a finite $\mathbb{Z}$-set $P$ to $A_n$ by sending every point to the length of its orbit modulo $n$. Equivariant injections preserve orbit lengths, so these maps form a cocone under the monomorphism subcategory; varying cyclic orbits shows that its legs are collectively surjective. If a local state classifier $L$ existed, its universal map onto every $A_n$ would be surjective because the universal legs are jointly epic. This would force the finite set $L$ to have at least $n$ elements for every $n$, a contradiction.

A natural transformation in a set-valued functor category is an epimorphism exactly when every component is a surjective function. More generally, limits and colimits in a functor category are computed pointwise whenever they exist in the codomain.

For small diagrams with complete codomain, choose each objectwise categorical limit. For $u:d\to d^{\prime }$, its projections uniquely define the arrow $L(u)$ through $p_j(d^{\prime })L(u)=H_j(u)p_j(d)$. The universal property proves identity and composition laws. The same projection calculation makes every categorical cone factorization a natural transformation. Hence these are genuine categorical limits in the functor category, and the objectwise forgetful functor is a limit-creating functor.

A natural transformation between functors to the Category of sets is a monomorphism in the functor category exactly when every component is an injective function. Componentwise cancellation proves sufficiency. For necessity, the Yoneda lemma converts two elements with equal component images into two transformations from a covariant representable functor; monicity cancels these transformations, forcing the elements to coincide. The argument remains valid for a locally small category in an ambient universe where the functor category is formed.

The presheaf category on a small category $\mathcal{C}$ is the functor category $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$. Its limits and colimits are computed pointwise.

Each object is a presheaf on a category, and its morphisms are natural transformations.

The right Kan extension along $K:\mathcal{C}\to \mathcal{D}$ is the right adjoint to precomposition by $K$ between suitable functor categories.

This is the right-adjoint form of Kan extension; left Kan extension is its left-adjoint counterpart.

In the covariant functor category $[\mathbb{N},\mathbf{Set}]$, the classifier at $n$ consists of thresholds $k\ge n$ and the empty threshold $\infty$. Its transition to $n+1$ sends $k$ to $\max (k,n+1)$. The displayed maps commute with transitions and are all surjective, so define an epimorphism. They identify $n$ and $n+1$, so the endomorphism is not an isomorphism.

An idempotent morphism $e:E\to E$ splits if there are $r:E\to H$ and $s:H\to E$ with $sr=e$ and $rs=1_H$. The object $H$ is a retract of $E$. In a functor category, the splitting maps must be natural transformations; objectwise splittings without compatible functorial choices are not enough.