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The Yoneda embedding is an exact representation of a category, but does not by itself reduce the subject to the study of arbitrary functor categories. There are both a size issue and a structure issue.

Let $\mathcal{A}$ be a locally small category and set $yA=\mathcal{A}(-,A)$. For a categorical presheaf $P:{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$, a natural transformation $\theta :yA\to P$ is determined by $x={\theta }_A(1_A)$. Indeed, naturality along $f:B\to A$ forcesConversely this formula defines a natural transformation for each $x\in P(A)$, since functoriality gives naturality along every $g:C\to B$. Taking $P=y{A}^{\prime }$ givesThus the Yoneda embedding is full and faithful, and $\mathcal{A}$ is equivalent to its full subcategory of representable presheaves. The indexing category in the usual version is ${\mathcal{A}}^{\mathrm{op}}$, not necessarily a small category chosen independently of $\mathcal{A}$.

Local smallness makes the values $\mathcal{A}(B,A)$ sets; it does not make the collection of objects a set. For a large indexing category, the collection of natural transformations between arbitrary set-valued functors can itself be a proper class. For example, take the discrete category on the class of ordinals and the constant two-element functor. Its endomorphisms include an independently chosen function $\{0,1\}\to \{0,1\}$ at each ordinal. A small functor category cannot encode this collection without changing universes. The Yoneda lemma remains meaningful when the source is representable because the displayed bijection identifies its natural transformations with a set. It does not make the entire large functor category locally small.

Even for a small category, the representable image is usually a very special part of its presheaf category. If $\mathcal{A}$ is the terminal category, its presheaf category is $\mathbf{Set}$, but its representable image consists only of a singleton. The presheaf category has all small categorical limits and colimits, whereas an arbitrary full subcategory need have almost none. The Yoneda embedding preserves existing limits because representable functors preserve limits. It generally does not preserve colimits. For example, the coproduct of two singleton sets in category of finite sets is a two-element set, but evaluating the proposed presheaf coproduct at a two-element set givesThe pointwise coproduct therefore leaves the representable image. Identifying which presheaves are representable, and which ambient constructions return to the original category, is precisely substantive category theory.

The representation is nonetheless powerful. The Yoneda lemma replaces a morphism by all its probes, converts a universal property into a representation of a functor, and makes the presheaf category a natural place to add formal colimits. A small presheaf category is the free free cocompletion under small colimits: every presheaf is a colimit of representables indexed by its category of elements. But this is a controlled enlargement of the category, not an identification of its original objects with all functors. The stronger conclusion in the proposed assertion therefore does not follow from the full-faithfulness statement.

A geometric embedding means $f_{\ast }$ is full and faithful, equivalently that the adjunction counit $f^{\ast }{f}_{\ast }H\to H$ is invertible for every $H$. At $c$, the explicit right-adjoint formula identifies this counit withinduced by ${\tau }_c:\mathbb{C}(-,c)\to \mathbb{D}(T-,Tc)$, sending $h$ to $Th$.

If $T$ is full and faithful, every ${\tau }_c$ is an isomorphism, making all these counit components bijective and hence making $f_{\ast }$ full and faithful. Conversely, if precomposition with ${\tau }_c$ is a bijection for every $H$, then ${\tau }_c$ is invertible by the Yoneda lemma applied in the presheaf category: taking $H=\mathbb{C}(-,c)$ gives a left inverse, and injectivity for $H=\mathbb{D}(T-,Tc)$ makes it a right inverse too. Evaluating ${\tau }_c$ at $c^{\prime }$ therefore says thatis bijective for all $c,c^{\prime }$. This proves the full-faithfulness criterion for presheaf geometric embeddings:

Write $\hat{\mathcal{C}}=[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and similarly for $\mathcal{D}$. The induced inverse-image functor is precomposition:All categorical limits in a presheaf category are computed pointwise. Thus $u^{\ast }$ preserves every limit, in particular finite limits.

Because the indexing categories are small and $\mathbf{Set}$ is complete and cocomplete, both Kan extensions along $f^{\mathrm{op}}$ exist. Their universal properties giveFor example their values are the comma-category formulasThe comma categories here are formed in ${\mathcal{D}}^{\mathrm{op}}$: an object of the first involves $f(c)\to d$ there, equivalently $d\to f(c)$ in $\mathcal{D}$. An object of the second involves $d\to f(c)$ there, equivalently $f(c)\to d$ in $\mathcal{D}$. Keeping the opposite category explicit prevents reversal of the two constructions.

The right-hand adjunction and finite-limit preservation are exactly the axioms for a geometric morphism $u:\hat{\mathcal{C}}\to \hat{\mathcal{D}}$. The extra left adjoint makes it an essential geometric morphism. On representable presheaves, its left adjoint satisfies $u_{!}(y_{\mathcal{C}}c)\cong y_{\mathcal{D}}(fc)$, by the Yoneda lemma and the left adjunction. This identifies the induced morphism directly with the original functor $f$.

For the covariant category in this part, put $\mathcal{E}=[\mathcal{C},\mathbf{Set}]$ and $r_c=\mathcal{C}(c,-)$. Its global-sections functor is $\Gamma =\mathrm{Hom}(1,-)$, the limit of the diagram. If $\Gamma$ is also an inverse image, it is a left adjoint and preserves all small colimits.

Consider the canonical pointwise-surjective mapAt $d$, the summand $r_d(d)$ contains $1_d$, so $q$ is an epimorphism. In a presheaf topos epimorphisms are regular, and a colimit-preserving functor preserves their coequalizer presentations. Thus $\Gamma q$ is surjective. Since $\Gamma$ preserves coproducts, some summand has a global element, yielding a section $s:1\to r_c$ of its unique map $r_c\to 1$. The constant singleton functor is a retract of a covariant representable.

Conversely, if $1$ is such a retract in a category, $\mathrm{Hom}(1,-)$ is a retract of $\mathrm{Hom}(r_c,-)$, which is evaluation at $c$ by the Yoneda lemma. Evaluation preserves colimits pointwise. The colimit comparison for a retract functor is a retract of the evaluation comparison; a retract of an isomorphism is an isomorphism, so $\Gamma$ preserves all small colimits. It also preserves finite limits because it is representable. A right adjoint can be constructed byFor $h:d\to d^{\prime }$, precomposition gives $r_{d^{\prime }}\to r_d$, and applying $\Gamma$ and then $\mathrm{Set}(-,S)$ gives $R(S)(d)\to R(S)(d^{\prime })$. The presentation of every functor as a colimit of representables gives $\mathrm{Hom}(F,R(S))\cong \mathrm{Set}(\Gamma F,S)$. Hence $\Gamma ⊣R$, so $\Gamma$ is an inverse image and the topos is local.

The retract condition has a precise interpretation as the initial object criterion for a covariant local topos. Write the section as a natural family $u_d:c\to d$. Naturality says $h{u}_d=u_{d^{\prime }}$ for every $h:d\to d^{\prime }$. Put $e=u_c$. Then $e^2=e$ and every $f:c\to d$ satisfies $fe=u_d$. In the idempotent completion of $\mathcal{C}$, the object $(c,e)$ is initial: an arrow from it to $(d,k)$ must satisfy $v=kve$, and the identities force $v=u_d$; this arrow exists because $k{u}_d=u_d$ and $u_{d}e=u_d$.

Conversely an initial object $(c,e)$ in the idempotent completion gives that natural family and the retract. ThereforeIf $\mathcal{C}$ is already idempotent-complete, this is equivalent to an initial object in $\mathcal{C}$ itself, and $\Gamma$ is evaluation there. The word is initial, not terminal, because this question uses covariant functors.

One cannot omit idempotent completion for a general small category. Take the one-object category of the monoid $\{1,0\}$ with absorbing zero. It has no initial object because its endomorphism set has two elements, but the natural family $u=0$ satisfies $m0=0$ for every $m$. Its covariant functor category is local. Concretely, a functor is a set with an idempotent operator, and its global sections are the fixed points of that operator.

For a topological space $X$, there is an equally explicit criterion. If $\mathbf{Sh}(X)$ is local, apply $\Gamma$ to the epimorphism ${\coprod }_i{U}_i\to 1$ associated with any open cover $X={\bigcup }_i{U}_i$, regarding the opens as subterminal sheaves. Since $\Gamma U_i$ is a singleton exactly when $U_i=X$, and otherwise empty, coproduct and epimorphism preservation force some $U_i=X$.

Thus every open cover of $X$ contains $X$ itself. Conversely this open cover criterion for a local sheaf topos condition is equivalent to existence of a point $x$ whose only open neighborhood is $X$: the union of all proper open sets cannot be all of $X$, so choose $x$ outside it. Such a point plainly forces any cover to contain $X$. Empty spaces fail the condition.

At this point, the stalk functor $F\mapsto F_x$ is simply $F(X)=\Gamma F$, since there is only one open neighborhood. Its right adjoint is the skyscraper sheaf of setsA morphism $F\to R(S)$ is determined exactly by its map $F(X)\to S$, because all proper opens omit $x$. Hence $\Gamma ⊣R$, and $\Gamma$ preserves finite limits, proving locality. ConsequentlyIn a $T_0$ space this distinguished point is unique and closed. In a $T_1$ space the condition forces $X$ to be a singleton. Indistinguishable points in a non-$T_0$ space can all have this property; uniqueness of an actual point is not part of the general criterion.

For the covariant version of the Yoneda embedding, put $Y(A)=\mathcal{C}(A,-)$. A morphism $u:A\to B$ induces the natural transformation $Y(u):Y(B)\to Y(A)$ whose component sends $v:B\to X$ to $vu$. Thus $Y$ is a functor from the opposite category ${\mathcal{C}}^{\mathrm{op}}$ to the functor category $[\mathcal{C},\mathbf{Set}]$. The locally small category hypothesis makes each value a set.

The Yoneda lemma says that, for a functor $F:\mathcal{C}\to \mathbf{Set}$, evaluation at the identity morphism gives a natural bijectionGiven $a\in F(A)$, define ${\theta }_X^a(v)=F(v)(a)$. For $w:X\to X^{\prime }$, the functor law gives $F(w){\theta }_X^a(v)=F(wv)(a)={\theta }_{X^{\prime }}^a(wv)$, proving naturality. Conversely, naturality of $\theta$ at $v:A\to X$ gives ${\theta }_X(v)=F(v){\theta }_A(1_A)$. These two constructions are inverse, because $F(1_A)(a)=a$.

The bijection is natural in both variables. A natural transformation $\beta :F\to H$ sends the element $a$ to ${\beta }_A(a)$, matching postcomposition by $\beta$. For $u:A\to B$, precomposing with $Y(u)$ sends $\theta$ to a transformation from $Y(B)$ and its distinguished element is ${\theta }_B(u)=F(u)(a)$. This is exactly the required dependence on $A$.

Taking $F=Y(B)$ gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$. Consequently the covariant Yoneda embedding is full and faithful, with the reversal of arrows accounted for by its opposite category domain.

Suppose every component ${\alpha }_A$ is a monomorphism in the Category of sets, hence an injective function. If $\beta ,\gamma :H\to F$ are natural transformations with $\alpha \beta =\alpha \gamma$, then ${\alpha }_A{\beta }_A={\alpha }_A{\gamma }_A$ at every object. Componentwise cancellation gives ${\beta }_A={\gamma }_A$, so $\beta =\gamma$. Thus $\alpha$ is a monomorphism in the functor category.

Conversely, suppose $\alpha$ is a monomorphism and $x,y\in F(A)$ satisfy ${\alpha }_A(x)={\alpha }_A(y)$. The Yoneda lemma supplies natural transformations ${\theta }^x,{\theta }^y:Y(A)\to F$. Its naturality in $F$ identifies $\alpha {\theta }^x$ and $\alpha {\theta }^y$ with the equal elements ${\alpha }_A(x)$ and ${\alpha }_A(y)$. Cancellation by $\alpha$ gives ${\theta }^x={\theta }^y$; evaluating at $1_A$ gives $x=y$. Therefore a natural transformation is a monomorphism exactly when all its components are injective functions. This is the pointwise monomorphism in a set-valued functor category criterion.

Use the covariant Yoneda embedding $Y(A)=\mathcal{C}(A,-)$. An arrow $f:B\to A$, viewed as an arrow $A\to B$ in the opposite category, induces precomposition $h\mapsto hf$. Identities and composition are preserved because composition in $\mathcal{C}$ is associative.

The covariant Yoneda lemma is the bijectionIts inverse sends $x\in F(A)$ to ${\alpha }_B(h)=F(h)(x)$. This is a natural transformation: for $k:B\to C$, functoriality gives $F(k){\alpha }_B(h)=F(kh)(x)={\alpha }_C(kh)$. Conversely, naturality of an arbitrary $\alpha$ at $h:A\to B$ gives ${\alpha }_B(h)=F(h){\alpha }_A(1_A)$. Evaluation at the identity morphism therefore makes the two constructions inverse. Their formulas also prove naturality in $F$ and, contravariantly, in $A$.

Taking $F=Y(B)$ identifies natural transformations $Y(A)\to Y(B)$ with $\mathcal{C}(B,A)$. Thus $Y$ is full and faithful, and reflects isomorphisms; two image objects are isomorphic exactly when the original objects are isomorphic. It also carries existing colimits in $\mathcal{C}$ to categorical limits of covariant representable functors: maps out of a colimit are exactly compatible families of maps out of its diagram. Equivalently, this embedding preserves existing categorical limits in ${\mathcal{C}}^{\mathrm{op}}$. The direction matters: the paper uses covariant representables, rather than the more usual contravariant Yoneda embedding.

The covariant form of the Yoneda lemma says that for a locally small category $\mathcal{C}$, a functor $F:\mathcal{C}\to \mathbf{Set}$ and an object $A$, there is a bijection, natural in $A$ and $F$,Explicitly its two directions areFor $u:B\to B^{\prime }$, functoriality gives $F(u){\alpha }_B^x(f)=F(uf)(x)={\alpha }_{B^{\prime }}^x(uf)$, so ${\alpha }^x$ is a natural transformation. Conversely, naturality of $\alpha$ at $f:A\to B$ gives ${\alpha }_B(f)=F(f)({\alpha }_A(1_A))$. This proves that the displayed maps are inverses.

Naturality in $F$ follows because postcomposition by $\theta :F\Rightarrow F^{\prime }$ sends $x$ to ${\theta }_A(x)$. For $u:A^{\prime }\to A$, precomposition of transformations by $\mathcal{C}(u,-)$ sends ${\alpha }^x$ to ${\alpha }^{F(u)(x)}$. This also verifies naturality in the representing object.

Take an epimorphism $\alpha :F\Rightarrow G$ in $[\mathcal{C},\mathbf{Set}]$ and a natural transformation $\beta :\mathcal{C}(A,-)\Rightarrow G$. By the Yoneda lemma, $\beta$ corresponds to $x={\beta }_A(1_A)\in G(A)$. By the pointwise epimorphism in a functor category criterion, choose $y\in F(A)$ with ${\alpha }_A(y)=x$.

The Yoneda lemma gives ${\gamma }_B(f)=F(f)(y)$, a natural transformation $\mathcal{C}(A,-)\Rightarrow F$. Naturality of $\alpha$ yieldsTherefore covariant representables are projective in the set-valued functor category. Local smallness ensures that $\mathcal{C}(A,-)$ is set-valued.

The key fact is that the extra left adjoint $f_{!}$ sends each representable $yC$ to an indecomposable projective object. Let ${\coprod }_i{B}_i\twoheadrightarrow f_{!}yC$ be epic. The inverse image $f^{\ast }$ preserves epimorphisms and coproducts, because it is a left adjoint between toposes. Apply it and lift the unit $\eta :yC\to f^{\ast }{f}_{!}yC$ through the resulting epimorphism, using projectivity of $yC$. A map from $yC$ to a coproduct selects one component, by evaluation at $C$ and the Yoneda lemma. Thus for some $i$ we obtain $t:yC\to f^{\ast }{B}_i$ with $f^{\ast }(B_i\to f_{!}yC)t=\eta$.

Transpose $t$ across $f_{!}⊣f^{\ast }$ to $\bar{t}:f_{!}yC\to B_i$. The displayed equality says that its composite back to $f_{!}yC$ is the identity. This proves the required indecomposable-projective property.

Since idempotents split in $\mathcal{D}$, part (ii) supplies objects $FC\in \mathcal{D}$ and isomorphisms $f_{!}yC\cong y(FC)$. Full faithfulness of the Yoneda embedding transports the action of $f_{!}$ on representable arrows to a functor $F:\mathcal{C}\to \mathcal{D}$. For $P\in \hat{\mathcal{D}}$,These identifications are natural in both $C$ and $P$. HenceIts right adjoint is consequently the right Kan extension from part (i), uniquely up to natural isomorphism. Thus the entire geometric morphism is induced by $F$.

A representable functor $yC$ is an indecomposable projective object. Given an epimorphism ${\coprod }_i{B}_i\twoheadrightarrow yC$, evaluate at $C$. Epimorphisms and coproducts in a presheaf category are pointwise, so $1_C\in yC(C)$ is the image of some element of a particular $B_i(C)$. By the Yoneda lemma, that element defines $s:yC\to B_i$, and its composite into $yC$ corresponds to $1_C$, hence is the identity. The selected component is split epic. More generally, evaluation sends any epimorphism to a surjection, so a map from $yC$ lifts through any epimorphism; this also proves its ordinary projectivity.

Conversely, every presheaf $A$ has the canonical epimorphismwhose component is the natural transformation named by $x$. It is pointwise surjective, since an element at $D$ is reached from its own summand $(D,x)$ at $1_D$. If $A$ is indecomposable projective, one component $r:yC\to A$ has a section $s:A\to yC$. The endomorphism $sr$ of $yC$ is idempotent and therefore corresponds to an idempotent morphism $e:C\to C$.

If idempotents split in $\mathcal{C}$, choose $C\stackrel{p}{\to }D\stackrel{i}{\to }C$ with $ip=e$, $pi=1_D$. Then $A\cong yD$: the mutually inverse maps are $y(p)s:A\to yD$ and $ry(i):yD\to A$. ThusWithout that hypothesis the argument still proves that every such object is a retract of a representable. The initial presheaf is not indecomposable projective, since its identity is the empty-coproduct epimorphism and has no component to select.

Write $\hat{\mathcal{C}}=[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and similarly for $\mathcal{D}$. The geometric morphism induced by a functor hasPrecomposition preserves all pointwise limits and colimits, so in particular it preserves finite limits. The Right Kan extension exists because the categories are small and sets have all small limits; its universal property gives $f^{\ast }⊣f_{\ast }$. Thus these functors define a geometric morphism $\hat{\mathcal{C}}\to \hat{\mathcal{D}}$.

There is also $f_{!}={\mathrm{Lan}}_{F^{\mathrm{op}}}$, a left Kan extension, with $f_{!}⊣f^{\ast }$. The Yoneda lemma identifies $f_{!}(yC)\cong y(FC)$, since for every $P$,This is the representable calculation used in the next parts.

For a locally small category $\mathcal{C}$, an object $C$, and a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$, the Yoneda lemma gives the natural bijectionIts two maps are explicitlyFor $u:A^{\prime }\to A$, the equation $X(u)X(f)(x)=X(fu)(x)$ proves naturality of $\Psi (x)$. Evaluating it at the identity morphism gives $\Phi \Psi (x)=x$. Conversely, naturality of $\alpha$ at $f:A\to C$ givesso $\Psi \Phi (\alpha )=\alpha$. Evaluation at the identity and transport of an element along a morphism are mutually inverse.

The bijection is natural in both variables: a natural transformation $\theta :X\Rightarrow Y$ sends $\Phi (\alpha )$ to ${\theta }_C(\Phi (\alpha ))$, matching $\Phi (\theta \alpha )$; and $g:C\to C^{\prime }$ givesFor completeness, the covariant Yoneda lemma for $Y:\mathcal{C}\to \mathbf{Set}$ is $\mathrm{Nat}(\mathcal{C}(C,-),Y)\cong Y(C)$, with $\alpha \mapsto {\alpha }_C(1_C)$ and inverse $y\mapsto (f:C\to A\mapsto Y(f)(y))$.

For the categorical presheaf $X$, its category of elements has objects $(C,x)$ with $x\in X(C)$. A morphism $(C,x)\to (D,y)$ is a morphism $f:C\to D$ satisfying $X(f)(y)=x$. Composition in a category is inherited from $\mathcal{C}$: if also $X(g)(z)=y$, then $X(gf)(z)=X(f)X(g)(z)=x$. The identity morphisms are inherited as well. The forgetful functor sends $(C,x)$ to $C$ and $f$ to $f$.

A universal element is a pair $(R,r)$ for which each $x\in X(C)$ is uniquely of the form $X(f)(r)$ for $f:C\to R$. Thus $(R,r)$ is a terminal object of the category of elements, with the variance appropriate to a categorical presheaf.

Given a universal element, defineThe defining uniqueness makes each map a bijection; $X(u){\psi }_C(f)={\psi }_{C^{\prime }}(fu)$ gives naturality for $u:C^{\prime }\to C$. Hence $\psi$ is a natural isomorphism and $X$ is a representable presheaf. Conversely, from a natural isomorphism $\psi :\mathcal{C}(-,R)\cong X$, take $r={\psi }_R(1_R)$. The Yoneda lemma gives ${\psi }_C(f)=X(f)(r)$; its bijectivity makes $(R,r)$ a universal element. Therefore the two descriptions coincide:

For $b:B\to B^{\prime }$ in $\mathcal{B}$, conjugate the induced natural transformation by the chosen representations of a functor:The Yoneda lemma gives a unique morphism $Gb:GB\to G{B}^{\prime }$ whose postcomposition map is ${\tau }_b$. Explicitly,Consequently, for every $f:A\to GB$,This is precisely the required naturality in $B$. The identities ${\tau }_{1_B}=1$ and ${\tau }_{b^{\prime }b}={\tau }_{b^{\prime }}{\tau }_b$ force $G{1}_B=1_{GB}$ and $G(b^{\prime }b)=G{b}^{\prime }Gb$, because the Yoneda embedding is fully faithful. Thus the representing objects extend uniquely to a functor $G:\mathcal{B}\to \mathcal{A}$. Uniqueness follows by the same Yoneda lemma: any candidate satisfying that naturality must induce ${\tau }_b$, so must have exactly the displayed action on morphisms.

First address the unnumbered preliminaries. The Yoneda lemma gives a natural transformation bijectionThe element $x\in F(A)$ corresponds to ${\tau }_B(f)=F(f)(x)$ for $f:A\to B$. These bijections are natural in $A$ and $F$; the contravariant form replaces $\mathcal{C}(A,-)$ by $\mathcal{C}(-,A)$.

Let $E=(1\downarrow F)$ be the category of elements. Its objects are $(A,x)$ with $x\in F(A)$; an arrow $u:(A,x)\to (B,y)$ is $u:A\to B$ with $F(u)(x)=y$. Since $\mathcal{C}$ is a small category, $E$ is small. Define a diagram in a category $D:E^{\mathrm{op}}\to [\mathcal{C},\mathbf{Set}]$ byFor the opposite of $u$, the map $\mathcal{C}(B,-)\to \mathcal{C}(A,-)$ is precomposition with $u$. The Yoneda lemma gives a cocone under a diagram $D\to F$ whose $(A,x)$ component sends $f:A\to B$ to $F(f)(x)$.

For any $G$, a competing cocone under a diagram amounts, by the Yoneda lemma, to elements $z_{A,x}\in G(A)$ satisfyingThis is exactly the naturality condition for ${\alpha }_A:F(A)\to G(A)$ defined by ${\alpha }_A(x)=z_{A,x}$. It gives a unique natural transformation $\alpha :F\to G$ factoring the cocone. Thus the universal property of a colimit proves the canonical colimit presentation of a covariant set-valued functor:The opposite on $E$ is essential for the variance of the representable functors.

We now prove the four conditions equivalent by the cycle $(i)\Rightarrow (ii)\Rightarrow (iii)\Rightarrow (iv)\Rightarrow (i)$. Suppose $F$ is a finite-limit-preserving set-valued functor. For a finite diagram in a category in the comma category $(S\downarrow F)$, write its objects as $(A_i,s_i:S\to F(A_i))$. Take $L={\lim }_i{A}_i$ in $\mathcal{C}$. Since $F$ preserves this finite limit, the compatible maps $s_i$ determine a unique $s:S\to F(L)$ whose composites with $F(L)\to F(A_i)$ are $s_i$. The pair $(L,s)$ has the required categorical limit property in $(S\downarrow F)$, by the same universal properties. For the empty diagram, $F(1)$ is a singleton, so there is exactly one map $S\to F(1)$; this provides the terminal object. Hence $(i)\Rightarrow (ii)$ for every set $S$.