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?