Framework
Let \(K\) be a field, \(\mathcal{C}\) a \(K\)-linear abelian category with finite dimensional ext groups. What we call a universal extension of object \(V\) by \(W\) is either an extension of \(V\otimes_K \mathrm{Ext}^1(V,W)\) by \(W\) or an extension of \(V\) by \(\mathrm{Ext}^1(V,W)^*\otimes_K W\), all of which corresponds to the identity morphism in \(\mathrm{End}(\mathrm{Ext}^1(V,W))\),
\[ \mathrm{Ext}^1(V\otimes_K \mathrm{Ext}^1(V,W),W) \cong \mathrm{Ext}^1(V,\mathrm{Ext}^1(V,W)^*\otimes_K W) \cong \mathrm{End}(\mathrm{Ext}^1(V,W)). \]
Denote the image of identity in the first and second group \(E_1(V,W)\) and \(E_2(V,W)\) respectively. The universal extension of \(V\) by \(W\) is then the universal object in \(\mathcal{C}\) such that for any extension
\[ 0 \to W \to X \to V \to 0 \]
there are commutative diagrams
and