I'm reading up on fine moduli spaces and I'm having difficulty seeing how every family over a scheme $B$ is the pullback of the universal family along a unique morphism. In fact, I'm not sure what this means.
To make my question more precise, I'll use the notation of Harris and Morrison, If $F$ is a moduli functor representable by a scheme $M$, let $\Psi: Mor(-,M) \to F$ be the corresponding natural isomorphism. Pulling back the identity on $M$, $1_M$, we get a family in $F(M)$, $\mathbf{1}: U\to M$. Let $\phi: D\to B$ be a family in $F(B)$. How does one realize this family as the pullback of $U$ via $\Psi(\phi)$?
Furthermore, I've seen the claim that $D\cong B\times_M U$. Harris and Morrison claim that there is a fibre product diagram
$\begin{array}{ccc} D &\rightarrow& U \\ \phi \downarrow && \downarrow \mathbf{1} \\ B&\xrightarrow{\Psi(\phi)}& M\end{array}$
but what is the top morphism and why is this a fibre product?
Best Answer
As pointed out in the comments, a functor acts not only on objects but also on morphisms. So in order to define your functor $F$, you have to specify not only a set $F(B)$ for each scheme $B$, but also a map $F(\varphi): F(B') \to F(B)$ for each morphism $\varphi: B \to B'$ of schemes. (This is assuming $F$ is contravariant.)
When $F$ is a moduli functor so that $F(B)$ classifies objects over $B$ up to some equivalence, this leads naturally to the fiber product. In order to specify $F(\varphi)$, for each class of objects $[X]$ over $B'$, we must specify some class of objects over $B$. In other words, we have the diagram
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that we need to complete. The instinctive choice is the fiber product $X \times_{B'} B$, and so we define $F(\varphi)([X]) = [X \times_{B'} B]$.
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As pointed out in the comments, the proof that every class of objects can be realized as the pullback of the universal object is the diagram chase in the proof of Yoneda's lemma. Let $h_M = \operatorname{Mor}(\, \cdot \,, M)$ and, using your notation, let $\Psi: h_M \overset{\sim}{\longrightarrow} F$ be the natural isomorphism. Given a scheme $B$ and an object $X \in F(B)$, then there exists a unique $\varphi \in h_M(B) = \operatorname{Mor}(B,M)$ such that $\Psi_B(X) = \varphi$. Let $\DeclareMathOperator{\id}{id} U = \Psi_M(\id_M)$ be the universal object over $M$. The very trivial fact that $\id_M \circ \varphi = \varphi$ then shows via the following diagram chase that $X$ is the pullback of $U$ along $\varphi$.
I haven't read Harris and Morrison's book carefully, but it looks like they breeze through the background of moduli problems. A more detailed resource that provides a careful treatment of moduli problems is Dinamo Djounvouna's masters thesis, The Construction of Moduli Spaces and Geometric Invariant Theory.