Let $(E,P_0)$ be an elliptic curve. Then I want to show that there is a short exact sequence associated to $E$; $$ 0\rightarrow p_1^* \operatorname{Pic}(E)\oplus p_2^* \operatorname{Pic}(E)\rightarrow \operatorname{Pic}(E\times E)\rightarrow \operatorname{End}(E,P_0)\rightarrow 0. $$ Clearly, one can find the injective map on the left hand side. However, I am having a bit of trouble figuring out how to show the cokernel is indeed the endomorphism group fixing $P_0$.
I've tried restricting the line bundle on $E\times E$ to the diagonal, to a component $\{P_0\}\times E$ and using the the fact that the Jacobian is $E$, and so forth. None of these seem to give me an obvious endomorphism which means pullbacks of line bundles on one component the trivial map. Any hints?