Representability of diagonal of $\mathscr{M}_g$

Let $\mathscr{M}_g$ be the moduli stack of genus $g$ curves ($g \geq 2$). That is, $\mathscr{M}_g$ is the category whose objects are proper smooth morphisms $f: C \to S$ whose geometric fibers are connected, genus $g$ curves. A morphism of curves $(f: C \to S)$ to $(f' : C' \to S')$ is an appropriate cartesian diagram.

Where can I find a reference containing a proof of the fact that the diagonal $\Delta : \mathscr{M}_g \to \mathscr{M}_g \times \mathscr{M}_g$ is representable?

In most places I've seen (for example in Edidin's notes), they prove representability of the diagonal by showing that $\mathscr{M}_g$ is a quotient stack . However, can one prove representability by hand? I know for instance that it boils down to the fact that if $C_1 \to S$ and $C_2 \to S$ are two curves, then we want that the functor $\text{Isom}_S(C_1,C_2)$ be representable by a scheme.

Suppose given $T \to S,$ and consider the pull-backs $C_{1/T}$ and $C_{2/T}$. If these are isomorphic, then the graph of the isomorphism will be a certain closed subscheme in $C_{1/T} \times_T C_{2/T}$ (which is the pull-back under $T \to S$ of the fibre product $C_1 \times_S C_2$) which is flat over $T$.
So Isom_S(C_1,C_2) is a certain locally closed subscheme of the Hilbert scheme of $C_1\times_S C_2$ over $S$. (To verify this, you have to think about what the conditions are on a closed subscheme of $C{1/T}\times_T C_{2/T}$ which is flat over $T$ for it to actually be the graph of an isomorphism.)