Given a category ${\cal C}$, a functor $F:{\cal C}\rightarrow {\rm Sets}$, 
an object $A$ in ${\cal C}$ and an element $x\in F(A)$, one may consider the smallest 
subfunctor $F\_x$ of $F$ that contains $x$. Explicitly,
$F\_x(B) = \{ F(f)(x) \}\_{f\in {\rm Hom}(A,B)}$ and 
$F\_x(f) = F(f)|\_{F\_x(B)}$ for $f:B\rightarrow B'$. 


While functors of the form
$F\_x$ are not necessarily representable, I believe a bit of the Yoneda lemma
survives. Namely given any set valued functor $G:{\cal C}\rightarrow {\rm Sets}$, 
and natural transformations $T,T'$ taking $F\_x$ to $G$,
$(T(id\_A))(x)= (T'(id\_A))(x)$ (in $G(A)$) implies $T=T'$. Thus one
can still identify the set of all natural transformations from $F\_x$ to $G$
with a subset of $G(A)$. The proof works just like Yoneda.


Can one give an intrinsic characterization of functors of the form $F\_x$ 
solely in terms of ${\cal C}$?



A weak Yoneda-type lemma for certain nonrepresentable functors?
Up to isomorphism, they are precisely quotients of representables. Indeed, these $F\_x$ arise as image (epi-mono) factorizations of the classifying map $\theta\_x: \hom(A, -) \to F$ of $x \in F(A)$: 


$$\theta\_x = (\hom(A, -) \stackrel{epi}{\to} F\_x \stackrel{mono}{\to} F)$$ 


since the image of the map $\theta\_x$ is by definition the smallest subobject of $F$ through which it factors. (Note that epi-mono factorizations in the presheaf topos are computed pointwise. Note also that if $Q$ is a quotient of a representable, $\theta: \hom(A, -) \to Q$, then $Q \cong Q\_x$ where $x \in Q(A)$ is the element $\theta\_A(1\_A)$.) 


This explains your modified Yoneda lemma in terms of the usual Yoneda lemma: to say $\theta: \hom(A, -) \to Q$ is epi is to say that for any two $T, T': Q \to G$, that $T \circ \theta = T' \circ \theta$ implies $T = T'$. But following the Yoneda lemma, $T \circ \theta$ is the unique map classified by the element $(T \circ \theta)\_A(1\_A)$, and your result follows. 


**Edit:** In my comment below in response to David, I referred to the end calculation of the set of natural transformations $F \to G$ between functors $F, G: C \to Set$ (where $C$ is small). In terms of vanilla products and equalizers, this set $Nat(F, G)$ is the equalizer of a pair of maps of the form 


$$\prod\_{A \in Ob(C)} G(A)^{F(A)} \stackrel{\to}{\to} \prod\_{A, B} G(B)^{F(A) \times \hom(A, B)}$$ 


I am almost tempted to leave the description of these maps to the reader. One takes a tuple of maps $\theta\_A: F(A) \to G(A)$ to the tuple whose component at $(A, B)$ is the evident composite $F(A) \times \hom(A, B) \to F(B) \to G(B)$; the other takes the same tuple to the evident composite $F(A) \times \hom(A, B) \to G(A) \times \hom(A, B) \to G(B)$. The condition that the maps are equalized corresponds exactly to the naturality condition on $\theta\_A: F(A) \to G(A)$. 


A nice exercise is to see the definition of sheaf over a space in terms of this construction. Let $U\_i$ be an open cover of $U$, and let $F = \bigcup\_i \hom(-, U\_i) \hookrightarrow \hom(-, U)$ be the corresponding covering sieve. Then a presheaf $G$ is a sheaf iff the restriction map 


$$G(U) \cong Nat(\hom(-, U), G) \to Nat(F, G)$$ 


is an isomorphism for every such covering sieve. 


**Edit 2:** This answer was downvoted for unspecified reasons. If there are good reasons, they should be made explicit. 


