Chinese remainder theorem is extremely important in the theory of rings, it is stated that there is a canonical isomorphism between $R/\bigcap I_i$ and $\prod R/I_i$. However, from the proof, neither did I see the reason for canonical, nor feel the 'beauty' of this theorem. That's to say, the proof doesn't adopt any categorical method. What I saw is checking the element and do the induction. I'd like a proof more advanced.
I found this question, however I'm not satisfied with this, since I don't understand why $L$, clearly not some sort of adjoint, keeps pullback, which is a type of limit.
Someone told me that is more or less nature from the sheaf perspective and $\operatorname{Spec}$. Nevertheless I believe that a categorical description will be more effective and clean.
I change my words in order to make myself more clear.
Ideals are just subobjects of $R$ in the category of $R$-modules (and $R$ itself is just the representing object for the forgetful functor to Abelian groups or to sets and can be described categorically).
This theorem is saying if you have a finite family of subobjects of $R$ which, pairwise, have categorical union (pushout from their intersection) $R$, then the canonical map $R\to\prod R/I$ induces an canonical isomorphism between $R$ quotient the intersection (pullback from $R$) of these subobjects - which is just $R/\bigcap I$ - and $\prod R/I$. Quotienting itself is just taking a certain coequaliser and is categorical.
This was phrased entirely element-free and in universal properties alone.