Chinese remainder theorem in abelian categories

Question

The chinese remainder theorem holds in arbitrary abelian categories? I found a generalization in homological categories, but i'm looking for a proof in valid in an abelian category.

Answer 1

I find for myself the following proof: we can obtain obtain the chinese theorem by applying nine lemma to the following diagram:

$\begin {matrix} &0&0&0&\\ &\downarrow&\downarrow&\downarrow&\\ 0\to&H\cap K\to&H\oplus K\to&H+K\to&0\\ &\downarrow&\downarrow&\downarrow&\\ 0\to&G\to&G\oplus G\to&G\to&0\\ &\downarrow&\downarrow&\downarrow&\\ 0\to&G/ H\cap K\to&G/H\oplus G/K\to&G/H+K\to&0\\ &\downarrow&\downarrow&\downarrow&\\ &0&0&0&\\ \end {matrix} $