Isom-functor for generalized elliptic curves is representable

Question

I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,

(page DeRa-61) (*) For $C_i$, generalized elliptic curves over $S$, with fibers smooth or $n$-gons, $n$ fixed, let $H_i \subset C_i^{\text{reg}}$ be subgroups isomorphic to $\underline{\mathbb Z/n\mathbb Z}$ which meet each irreducible component of each geometric fibers ($i = 1,2$), $\underline{\operatorname{Isom}}((C_1, H_1, +), (C_2, H_2, +))$ is representable. According to II 1.17, the morphism $$\underline{\operatorname{Isom}}((C_1, H_1, +), (C_2, H_2, +))\to\underline{\operatorname{Isom}}((C_1/H_1, e), (C_2/H_2, e))$$ is representable by an étale morphism, while it results from the theory of Hilbert functor that $\underline{\operatorname{Isom}}((C_1/H_1, e), (C_2/H_2, e))$ is representable.

I am new to Isom-functors and Hilbert functors and I was looking for an explanation for why

1. $\underline{\operatorname{Isom}}((C_1, H_1, +), (C_2, H_2, +))\to\underline{\operatorname{Isom}}((C_1/H_1, e), (C_2/H_2, e)),$ and

2. $\underline{\operatorname{Isom}}((C_1/H_1, e), (C_2/H_2, e))$

are representable (by schemes).

For reference, here's the statement of Theorem II.1.17:

Theorem II.1.17 (DeRa-40). Let $X$ be a generalized elliptic curve over $S$ with a unit section $e$. Let $u : Y \to X$ be a finite étale covering of $X$ with a section $e \in Y(S)$ above $e.$ Suppose that the geometric fibers of $Y/S$ are connected. Then, there exists a unique structure of a generalized elliptic curve on $Y$ with unit $e$, such that the diagram \begin{matrix} Y^{\text{reg}} \times_S Y & \xrightarrow{+} & Y \\ \downarrow & & \downarrow{u} \\ X^{\text{reg}}\times_S Y & \xrightarrow{+} & X \end{matrix} is commutative.

Any help will be appreciated.