What's the intuition behind "representable morphisms"? A central notion in many algebro-geometrical stuff appears to be so-called "representable morphisms". A general (read: hand-wavy) definition could be the following, as far as I can tell:

Let $\mathsf{C}$ be a site, and let $f : F \to G$ be a morphisms of sheaves over $\mathsf{C}$. Then $f$ is representable by an object of type $T$ if, for all representable sheaves $h_X$ and all morphisms $h_X \to G$, the fiber product $F \times_G h_X$ is of type $T$. (Here the type $T$ could be "affine scheme", "scheme", "algebraic space", "stack", "manifold", "differentiable manifold", "open subset of $\mathbb{R}^n$"...)

I'm more used to topology, so of course my first instinct was to try and see what this meant for topological spaces. As far as I can tell, a continuous map $f : X \to Y$ is representable by an open subset of $\mathbb{R}^n$ (resp. a manifold) iff for all open subsets of $Y$ that are homeomorphic to an open subset of $\mathbb{R}^n$, then $f^{-1}(U)$ is an open subset of $X$ homeomorphic to an open subset of $\mathbb{R}^n$ (resp. to a manifold).
Honestly, this isn't very enlightening... I can follow the definition and understand how the "representability" condition is used, but I have no real intuition for it. What does it mean? Surely the name wasn't chosen randomly, so if a morphism is represented by (say) a scheme, then what scheme is that, and in what sense does it represent $f$?
 A: Let $\mathcal T$ be the sheaves of type T, for $X\in\mathsf C$ let  be $\mathcal T\downarrow X$ be the category of arrows $T\to h_X$ for $T\in\mathcal T$ and let $\mathcal G$ be the Grothendieck construction of $X\mapsto \mathcal T\downarrow X$. Elements of $\mathcal G$ are of the form $r=\{r_{T,X}\overset{r_{f,X}}\to h_X\}_{X\in\mathsf C}$.
Let $f:F\to G$ and let $R_f$ be the functor $\mathcal G\to\mathsf{Set}$ such that $R_f(r)$ is the set of all big commutative diagrams
$$
\newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!}
\newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.}
\begin{array}{ccc}
r_{T,X}&\ra{r_{f,X}}&h_X\\
\da{a_X} & & \da{b_X}\\
F\phantom{\text{ }}\phantom{\text{ }} & \ra{f} &G\phantom{\text{ }}\phantom{\text{ }}
\end{array}$$
indexed by $a$ and $b$, where $X$ varies over the site $\mathsf C$
Then, $f$ should be representable iff $R_f$ is representable in the sense of functors. Is that the kind of idea you were looking for ?
