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The terminal object $1$ is an exponentiable object because $-\times 1\cong 1_{\mathcal{C}}$ has the identity as a right adjoint. If $A$ and $B$ are exponentiable objects, compose the two adjunctions to getConsequently exponentiable objects are closed under finite products in a category, including the empty product, and

Under the additional hypotheses, a coseparator $S$ detects unequal parallel arrows by postcomposition: for $u\ne v:X\to B$ there is $t:B\to S$ with $tu\ne tv$. Local smallness makes $I=\mathcal{C}(B,S)$ a set, and completeness forms the product in a category $S^I$. The evaluation arrowis a monomorphism, because $e_{B}u=e_{B}v$ forces $tu=tv$ for every $t$, and the coseparator then forces $u=v$.

By hypothesis this monomorphism is a regular monomorphism, so it is an equalizer of some $r,s:S^I\rightrightarrows Q$. Apply the same evaluation construction to $Q$: $J=\mathcal{C}(Q,S)$ is a set and $e_Q:Q\to S^J$ is monic. Cancelling $e_Q$ shows that equalizing $e_{Q}r,e_{Q}s$ is equivalent to equalizing $r,s$. We have therefore proved the equalizer presentation by powers of a coseparator:The notation $S^I$ here means a product indexed by a set, so its existence does not presuppose categorical exponentiation.

If $A$ is an exponentiable object, $\mathcal{C}(-\times A,S)\cong \mathcal{C}(-,S^A)$ is a representable functor. Conversely, suppose a representing object $E$ and a natural isomorphismare given. For any set $I$, the product in a category property givesThus maps into all powers of $S$ have representations. For an equalizer presentation $B\to S^I\rightrightarrows S^J$ with arrows $r,s$, postcomposition induces natural transformations between these represented functors. The Yoneda lemma identifies them with arrows $r_A,s_A:E^I\rightrightarrows E^J$. Form their equalizer $B_A$. ThenThese bijections are natural in $X$. For a morphism $B\to C$, postcomposition and the Yoneda lemma give the unique arrow $B_A\to C_A$ respecting them; uniqueness proves functoriality. Hence $B\mapsto B_A$ is a right adjoint to $-\times A$. This proves the exponentiability test using a coseparator:

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

Assume (b). A coproduct in a category of monofunctors in $[\mathcal{C},\mathbf{Set}]$ is again a monofunctor: at an object it is the disjoint union of the values, and every arrow acts within the same summand by an injective function. Distinct summands cannot become identified, and injectivity within a summand follows from its monofunctor property.

Apply this to the coproduct of representable functors constructed in part (a). Its canonical map to $F$ is a pointwise epimorphism in a functor category. Thus every $F$ is an epimorphic image of a monofunctor, proving

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:

For a locally small category $\mathcal{C}$, an object $A$, and a functor $F:\mathcal{C}\to \mathbf{Set}$, the covariant Yoneda lemma is the natural bijectionIts inverse sends $x\in F(A)$ to the natural transformation whose component at $B$ maps $f:A\to B$ to $F(f)(x)$.

Suppose now that $\mathcal{C}$ is a small category. For $F:\mathcal{C}\to \mathbf{Set}$, form the coproduct in a categoryThe Yoneda lemma associates to every summand the natural transformation determined by $x$, and these transformations combine to a map $p:P\to F$. At an object $B$, the element $y\in F(B)$ is the image of $1_B$ in the summand indexed by $(B,y)$, so $p$ is a pointwise epimorphism in a functor category. Each representable functor $\mathcal{C}(A,-)$ is a projective object in a category, sinceand evaluation preserves pointwise epimorphisms. A coproduct of projectives is projective, so $P$ is the required projective object. This is the projective cover of a set-valued functor by representables.

We next prove the three equivalent conditions. If every morphism of $\mathcal{C}$ is a monomorphism, then for $f:B\to C$ and every $A$, postcompositionis injective. Thus every covariant representable functor is a monofunctor. Conversely, taking $A=B$ shows that injectivity for every representable implies that $fg=fh$ forces $g=h$, so every $f$ is monic.

If all representables are monofunctors, the object $P$ above is a monofunctor because a coproduct of injective functions is injective. Hence every $F$ is an epimorphic image of a monofunctor. Conversely, suppose every functor is an epimorphic image of a monofunctor and apply this to a representable $R$. Choose an epimorphism $q:M\twoheadrightarrow R$ with $M$ a monofunctor. Since $R$ is projective, $1_R$ lifts to $s:R\to M$ with $qs=1_R$. Thus $R$ is a retract in a category of $M$. Every retract of a monofunctor is a monofunctor: if $R(f)x=R(f)y$, then injectivity of $M(f)$ applied to $s(x)$ and $s(y)$ gives $s(x)=s(y)$, and applying $q$ gives $x=y$. This completes the equivalence.

Finally, every functor $\mathcal{C}\to \mathbf{Set}$ is a monofunctor exactly when every morphism of $\mathcal{C}$ is a split monomorphism. The forward implication is immediate because every functor preserves a left inverse. For the converse, fix $f:A\to B$ and form a quotient of $\mathcal{C}(A,-)\bigsqcup 1$ by identifying the distinguished point with every arrow of the form $kf:A\to X$, where $k:B\to X$. In the resulting functor $Q$, the two elements $[1_A]$ and $\ast$ of $Q(A)$ have equal images under $Q(f)$ because $[f]=\ast$. If every functor is a monofunctor, $Q(f)$ is injective, so $[1_A]=\ast$. By construction this means $1_A=rf$ for some $r:B\to A$. Thus $f$ is split monic. Equivalently, every morphism of $\mathcal{C}$ must be an absolute monomorphism.

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 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.

A functor between regular categories is regular if it preserves finite limits and regular epimorphisms. It then preserves image factorizations, since finite-limit preservation preserves monomorphisms. A representable functor $\mathcal{C}(P,-)$ is regular exactly when $P$ is projective with respect to regular epimorphisms. Regularity of a functor does not alone imply faithfulness or reflection of isomorphisms.

A sieve on $U$ is a subfunctor of the representable functor $yU$, equivalently a family of arrows into $U$ closed under precomposition. For $f:V\to U$, the pullback sieve consists of arrows $g$ into $V$ for which $fg$ belongs to the original sieve.

A Grothendieck topology is subcanonical when every representable functor is a sheaf on a site. It is also called standard in some topos-theory terminology. A nontrivial covering quotient can fail this condition by identifying distinct sections of a representable.