From presheaves to categories of presheaves I am trying to understand categories of presheaves, starting from a somewhat hazy understanding of presheaves.
An example is with some topological space X, where we define the category $ \mathcal Top(X) $ having as objects the open sets in X and with set inclusion as morphisms. The presheaf here is a contravariant functor into $ \mathcal Set $, taking each U in $ \mathcal Top(X) $ to the set (ring) of continuous maps from U to $ \Bbb R $, and inclusion of sets into restrictions of maps.
Now, to construct the category of preheaves over Top(X), I understand that we have to vary on what set each presheaf selects for each object in $ \mathcal Top(X) $, which to me means varying on the codomain of the maps that make up those sets; so each presheaf will take U in $ \mathcal Top(X) $ to the set of maps from U to ($ \Bbb R $ or some other alternative). What would these alternatives be?
With another example, thinking of the category of presheafs over some monoid, a presheaf woud be a functor into $ \mathcal Set $, thus hitting a single set - of maps from the monoid object (*) to what? Do I have to think of maps here at all? A representable presheaf would be isomorphic to $ \mathcal Hom( \_ , *) $; taking * to the set of morphisms (maps?) from * to itself; right? What other codomains would other presheafs use?
Edited to clarify: I understand that a presheaf is just a contravariant functor into $ \mathcal Set $, and that functoriality will make sure that the structure of $ \mathcal C $ is carried over by the presheaf. However, I'm trying to understand what exactly is this structure that is being carried over.
 A: This is basically wrapping up the discussion between me and OP in the comments, with some extra examples.
First of all, let's recall the definition of a presheaf.

Definition. A presheaf on a (small) category $\mathcal{C}$ is a contravariant functor into $\mathbf{Set}$. That is, a functor $P: \mathcal{C}^\text{op} \to \mathbf{Set}$.

In the question there are already explicit examples of presheaves, and the question is then if we can say something about the general structure of a presheaf. Unfortunately, we can in general not say much more than the definition already tells us.
There are lots of possible presheaves. For example, we can always take $P: \mathcal{C}^\text{op} \to \mathbf{Set}$ to send every object $C$ in $\mathcal{C}$ to the singleton $\{*\}$ and every arrow in $\mathcal{C}$ is then sent to the identity. This gives us a presheaf. More generally, for a set $X$ we can always define the constant $X$ presheaf: all objects are sent to $X$ and every arrow is sent to $Id_X$.
If we assume $\mathcal{C}$ to be of a particular form, then we can sometimes say something more about what (some of) the presheaves are. One example is already given in the question: take $\mathcal{C}$ to be the opens of a topological space $X$, and take $Y$ to be another topological space (in the question we have $Y = \mathbb{R}$). Then we have a presheaf sending an open $U$ to the continuous functions $U \to Y$ and $U \supseteq V$ (an arrow in $\mathcal{C}^\text{op}$) is sent to the restriction of those functions to $V$.
Another example is also mentioned in the question. If $\mathcal{C}$ is a monoid seen as a category, then a presheaf is actually just a set with a right monoid action. That is, $\mathcal{C}$ has one object $*$, an arrow for every element in the monoid and composition is given by the monoid operation. Then an arbitrary presheaf $P: \mathcal{C}^\text{op} \to \mathbf{Set}$ will have as data a set $P(*)$ and for every element $a$ of the monoid a function $P(a): P(*) \to P(*)$. Since $P$ is required to be a (contravariant) functor, we get that the monoid acts on the right on $P(*)$: for $x \in P(*)$ we take $xa$ to be $P(a)(x)$, then $x(aa') = P(aa')(x) = P(a')(P(a)(x)) = (xa)a'$.
In particular, if in the above $\mathcal{C}$ is a group, then the presheaves are just sets with a right group action (of that group).
Finally, there is one more important example that works for general (small) $\mathcal{C}$. Namely that of the representable functor. For every object $C$ in $\mathcal{C}$ we get a presheaf $\operatorname{Hom}(-, C): \mathcal{C}^\text{op} \to \mathbf{Set}$. That is, for $C'$ this just gives us $\operatorname{Hom}(C', C)$, the set of arrows $C' \to C$. For an arrow $f: C'' \to C'$, this gives us a function $\operatorname{Hom}(f, C): \operatorname{Hom}(C', C) \to \operatorname{Hom}(C'', C)$ by sending $g: C' \to C$ to $gf: C'' \to C$.
This last example is important because of the Yoneda lemma (nLab, wiki). It follows from the Yoneda lemma that we can find $\mathcal{C}$ as a full subcategory of $\mathbf{Set}^{\mathcal{C}^\text{op}}$, the category of presheaves on $\mathcal{C}$ (with natural transformations between them).
