This is a follow-up to the question here, but I think sufficiently new to merit a new question.
The answer to that question implies that defining a canonical reduction map between the isogenies from $E_1 / L$ to $E_2 / L$ and the isogenies between their reductions at some prime $\mathfrak{P}$ of $L$ is "messy at best" without using the Neron model. I don't doubt that the Neron model is exceptionally useful here, but I'm wondering why we can't use the proposition stated earlier (II.2.2) that if $E_1$ and $E_2$ are defined over $L$ then there exists a finite extension $L'$ of $L$ such that every isogeny from $E_1$ to $E_2$ is defined over $L'$?
Instead of using the Neron model, can we just modify the hypotheses of Proposition II.4.4 to say: Let $E_1 / L$ and $E_2 / L$ be elliptic curves with good reduction at some $\mathfrak{p}$ of $L$. Let $L'$ be a finite extension of $L$ such that $\operatorname{Hom}(E_1 , E_2)$ is defined over $L'$ and $E_1$, $E_2$ have good reduction at a prime $\mathfrak{P}$ of $L'$ lying over $\mathfrak{p}$.
We can then take the usual minimal Weierstrass models for $E_1, E_2$ over the completion of $L'$ at $\mathfrak{P}$, and scale all our isogenies so that their coefficients are in the ring of integers of $L'_{\mathfrak{P}}$ since our curves are projective varieites. Then it seems there is a "natural map" between $\operatorname{Hom}(E_1, E_2)$ and $\operatorname{Hom}(\tilde{E_1},\tilde{E_2})$ just by reduction of coefficients modulo $\mathfrak{P}$. Are there some subtleties in making this map that I'm missing?