Suppose I have a degree 2 line bundle $L$ on an elliptic curve $E$. Then we may write $L = \mathcal{O}(p+q)$ for some points $p,q \in E$. If we let $\oplus$ denote the addition map on the elliptic curve $E$. Then we can map $L$ to the degree 1 line bundle $L' = \mathcal{O}(p \oplus q)$.
Is this map well defined? Does it define a morphism from the moduli space of degree 2 line bundles on $E$ to the moduli space of degree 1 line bundles on $E$?