I am confused about the following. I know that for two line bundles $L_1, L_2$ on an algebraic curve $C$ the vector space ${\rm Ext}^1(L_1,L_2)$ classifies isomorphism classes of rank two vector bundles on $C$ which are extensions of $L_2$ by $L_1$. My question is, does this mean that there is a "universal" rank two vector bundle $E$ on ${\rm Ext}^1(L_1,L_2) \times C$ ?

I.e. if I let $p_2 : {\rm Ext}^1(L_1,L_2) \times C \to C$ be the projection then does there exist a rank two vector bundle $E$ on ${\rm Ext}^1(L_1,L_2) \times C$ such that $0 \to p_2^*L_1 \to E \to p_2^*L_2 \to 0 $ is a short exact sequence and for all $x \in {\rm Ext}^1(L_1,L_2)$ if we restrict the above short exact sequence to $x \times C$ we get the exact sequence $0 \to L_1 \to E_x \to L_2 \to 0 $ of vector bundles on $C$ which corresponds to the extension class $x$ ?