The non-existense of the fine moduli scheme of vector bundles. Why? The reference I am using is  Norbert Hoffmann's The moduli stack of vector bundles on a curve
. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a projective curve. Let $S$ be a $k$-scheme then we denote by
$$\text{Bun}_{r,d}(S) = \{ \mathcal{E} \text{ vector bundles on } C\times_k S \text{ rank $r$ and degree $d$ } \}/\backsim$$
the set of isomorphism classes of vector bundles $\mathcal{E}$ on $C \times_k S$. Now, every morphism of $k$-schemes $f:T\to S$ induces a pullback map
$$ f^* : \text{Bun}_{r,d}(S) \to \text{Bun}_{r,d}(T)  $$ with
$$ [\mathcal{E}] \mapsto [f^* \mathcal{E}]. $$
Question 1: Should it not be $$ f^* : \text{Bun}_{r,d}(T) \to \text{Bun}_{r,d}(S) \,?$$
Thus we get the contravariant functor
$$ \text{Bund}_{r,d}(-) : \text{Schemes over $k$} \to \text{Sets} $$
from the category of schemes to the category of sets. Then, we have the definition of the fine moduli scheme.
Definition A scheme $M$ over $k$ is a fine moduli scheme for vector bundles (of rank $r$ and degree $d$) on $C$ if $M$ represents the functor Bun$_{r,d}(-)$.
Question 2: What does this requirement actually mean? I.e. that a scheme represents a functor as above?
More explicitly, the author continues, and this is where I get confused mostly, $M$ is a fine moduli scheme of vector bundles if there exists the following functorial bijection:
$$ \{ \phi : S \to M \text{ a $k$-morphism} \} = \{ \mathcal{E} \text{ vect. bundle of rank $r$ and degree $d$ } \} /\backsim $$
Question 3 How exactly can I understand this equality? It is not quite clear what the objects a morphism in both sides are and why there is some isomorphism between them.
Finally, the whole point is to show that $M$ does not represent the functor Bund$_{r,d}(-)$ which actually is not representable (thus the need for the moduli stack). To show this the author uses the gluing example. In specific

*

*for any $k$-scheme $M$ a $k$-morphism $\phi : S \to M$ is given b a
$k$-morphism $\phi_i:U_i \to M$ such that in intersection $U_{ij}=U_i \cap U_j$ we have $\phi_i=\phi_j$.

*a vector bundle $\mathcal{E}$ over $C \times_k S$ is given by a vector bundle $\mathcal{E}_i :C \times_k U_i $, $U_i \subset \mathcal{E}$,for each $i$, an isomorphism $a_{il} = \mathcal{E}_i \to \mathcal{E}_j$ in the intersection, and the cocycle condition $a_{il} = a_{jl} \circ a_{ij}$ on triple intersections.

The author says that the these two objects behave completely differently under gluing but since I do not see their functorial isomorphism I do not see the authors point.
Question 4 Would you be able to clear this point out and explain it?
 A: I'm worried that I'm going to rewrite a Wikipedia article but let's try this. The general setup is this: one has a category $\mathscr{C}$ and a contravariant functor $F$ from $\mathscr{C}$ to the category of sets. $F$ is representable if there is an object $M$ of $\mathscr{C}$ and an isomorphism of functors $\Theta\colon h_M \to F$, where $h_M = \operatorname{Hom}_{\mathscr{C}}(-, M)$. Great.
Now, Yoneda tells you something interesting. The morphism $\Theta$ is determined by the element $\xi = \Theta_M(1_M)$ of $F(M)$. Why? One checks that for any object $X$ we're forced to have the bijection $\Theta_X\colon \operatorname{Hom}(X,M) \to F(X)$ given by $\Theta_X(f) = F(f)(\xi)$.
In your case, the requirement is that there is some scheme $M$ and a vector bundle $\mathscr{U}$ of rank $r$, degree $d$ on $M \times C$. So you also have to construct this bundle. From any morphism $f\colon X\to M$ we get a bundle $(f \times 1_C)^*\mathscr{U}$ on $X \times C$, and we have to get each bundle on $X \times C$ from a unique $f$.
