In Katz and Mazur's book "Arithmetic moduli of elliptic curves" (available here), two equivalent definitions of a $\Gamma _1(N)$-structure (point of exact order $N$) are given on page 99 (page 55 in the pdf). While I see why one implication holds, I fail to understand the other one. Let me sum up the statement.
Let $S$ be an arbitrary scheme and $E$ be an elliptic curve over $S$. Let $N\geq 1$ be an integer. Assume that we are given a group homomorphism $$\phi: \mathbb{Z}/N\mathbb{Z}\rightarrow E[N](S)$$ such that the effective Cartier divisor in $E$ $$G:=\sum_{a \operatorname{mod} N}[\phi(a)]$$ is a subgroup-scheme of $E$ (ie. $\phi(1)$ is a point of exact order $N$).
Then, there exists an elliptic curve $E'$ over $S$ together with an $N$-isogeny of elliptic curves over $S$ $$E\xrightarrow {\pi}E'$$ such that we have an equality of effective Cartier divisors in $E$ $$\operatorname{Ker}\pi = G$$
I fail to understand how one can define such a morphism $\pi$. Actually, I am not even sure about how to define a suitable elliptic curve $E'$. Because I want the kernel of my morphism to be the given subgroup-scheme induced by $\phi$, I would like $E'$ to be, in some sense, "$E/G$". However, I am not even sure about the good definition of such an object, and whether it is an elliptic curve (over $S$).
Would someone see what is the correct argument here? Could someone please explain to me how $E'$ and $\pi$ are defined with respect to $\phi$ and $G$?
I thank you very much for your help.