Recovering a topological space from one of its hom-functors An example often given to motivate the Yoneda Lemma is as follows.
Suppose $X$ is a topological space, and let $\{*\}$ and $S$ denote the singleton and Sierpinski topological spaces respectively.
Define a topological space $X'$ whose set of points is given by $\mathrm{Hom}(\{*\}, X)$, and for which $U \subseteq \mathrm{Hom}(\{*\}, X)$ is open iff, for some $f \in \mathrm{Hom}(X, S)$, we have
$$
f \circ g(*) = 1 \qquad \text{for all $g \in U$}
$$
Then it holds that $X'$ is homeomorphic to $X$. In other words, given access to both $\mathrm{Hom}(-, X)$ and $\mathrm{Hom}(X, -)$, we can recover $X$ up to isomorphism (in the form of $X'$).
I would like to find a similar construction that recovers $X$ up to isomorphism using only one of $\mathrm{Hom}(-, X)$ and $\mathrm{Hom}(X, -)$. It seems that this should be possible: we simply need to search over topological spaces until we find $Y$ such that $\mathrm{Hom}(-, Y)$ (or $\mathrm{Hom}(Y, -)$) is isomorphic to $\mathrm{Hom}(-, X)$ (or $\mathrm{Hom}(X, -)$) in the appropriate functor category, and then use the Yoneda Lemma to infer that $Y$ is isomorphic to $X$. However, I can't see an explicit construction in the spirit of the example above. What is one example?
 A: For any category $\mathcal{C}$ and any functor $F : \mathcal{C}^\textrm{op} \to \textbf{Set}$, we may form the following category $\textbf{El} (F)$:

*

*An object in $\textbf{El} (F)$ is a pair $(T, x)$ where $T$ is an object in $\mathcal{C}$ and $x$ is an element of $F (T)$.


*A morphism $f : (T', x') \to (T, x)$ in $\textbf{El} (F)$ is a morphism $f : T' \to T$ in $\mathcal{C}$ such that $F (f) (x) = x'$.


*Composition and identities are inherited from $\mathcal{C}$.
There is an evident forgetful functor $P : \textbf{El} (F) \to \mathcal{C}$.
Unfolding the definitions, we find that
$$\varprojlim \mathcal{C} (P, X) \cong [\mathcal{C}^\textrm{op}, \textbf{Set}] (F, h_X)$$
where $h_X = \mathcal{C} (-, X)$.
These bijections are natural in $X$, so we deduce that the colimit $\varinjlim P$ exists in $\mathcal{C}$ if and only if $F$ is representable, and moreover $\varinjlim P$ is a representing object for $F$.
Alternatively, we might observe that terminal objects in $\textbf{El} (F)$ are the same thing as representations of $F$.
(You might like to check for yourself that $(S, \textrm{id}_X)$ is a terminal object in $\textbf{El} (h_S)$.)
The colimit of a diagram with a terminal object is the value of the diagram at that object, so we also recover the earlier result.
This is by no means effective or practical, but at least it works for arbitrary categories.
A: In fact, the topological space $X$ can be recovered (up to homeomorphism) using just the functor $\operatorname{Hom}({-}, X)$.  To see this, as before, define $X' := \operatorname{Hom}(\{ * \}, X)$.
Now, for each nonempty directed set $I$, give $I \sqcup \{ \infty \}$ the topology where $U$ is open if and only if $\infty \in U \rightarrow \forall^{\infty} i \in I, i \in U$ (where $\forall^\infty$ is notation for the "eventually" quantifier, i.e. $\exists i_0 \in I, \forall i \in I, i \ge i_0 \rightarrow i \in U$).  It is then straightforward to show that this indeed defines a topology, and that for a net $x : I \to X$ and point $y \in X$, $x_i \to y$ if and only if the function $x \sqcup y : I \sqcup \{ \infty \} \to X$ is continuous.
Since the question of what nets converge to what points uniquely determines the topology, this shows that there is a unique topology on $X'$ such that a net $x' : I \to X'$ converges to a limit $f \in X'$ if and only if there exists $g \in \operatorname{Hom}(I \sqcup \{ \infty \}, X)$ such that $g \circ \operatorname{inc}_I = x'$ and $g \circ \operatorname{inc}_{\{\infty\}} = f \circ (\{ \infty \} \to \{ * \})$.  And furthermore, $X'$ is homeomorphic to $X$.
(This does have somewhat of the drawback that we are not using a small set of values of $\operatorname{Hom}({-}, X)$, whereas the original construction of course used only one value each from $\operatorname{Hom}({-}, X)$ and $\operatorname{Hom}(X, {-})$, along with the compositions of the two.  Though if you trace back the proof that net convergence uniquely determines topology, you can in fact reduce the collection of directed sets that you need to use to a small set, but still one that is dependent on $X$.  Also, if there is a nice description of what data of net convergence come exactly from a topology, I haven't seen it -- in contrast to the cases of e.g. closure operator, a basis of the space, a collection of neighborhood bases of the space, the data of what subsets are neighborhoods of what points, etc.)
