As the title says, I would like to understand an isomorphism of Hida families from a more geometric perspective than what I normally read. What bothers me is that there are two construction of the universal Hecke (ordinary) Hida-Hecke algebra and they turn out to give isomorphic objects: fix a prime $p\geq 5$, and a tame level $N$ prime to $p$.
Theorem 1.1 in the quoted paper by Hida shows that these two algebras are isomorphic (in the most compatible way one can dream of, in particular inducing the same Hecke action on spaces of cusp forms). Hida interpretes the space of cusp formsof weight $k$ and level $Np^r$ as global section of a sheaf of differentials ''twisted $k-2$ times'' over the closed modular curve $X_1(Np^r)$, as usual, ) but the limit process his proof is purely entirely algebraic- at least as far as I understand.