The reference I am using is this one. 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 (only because we have arrow reversal, but how do we know this? I.e. this is the same as Question 1) $$ \text{Bund}_{r,d}(-) : \text{Schemes over $k$} \to \text{Sets} $$ 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?

I asked this question on Math.StackExchange but nobody answered.