Let $X$ be a curve over an algebraically closed field $k$ (even over $k = \mathbb{C}$ if you want), let $J = Pic^0_{X/k}$ be its Jacobian, let $P \in X(k)$ be a point, and let $i \colon X \hookrightarrow J$ be the closed immersion that sends $Q$ to the divisor class of $[Q] - [P]$. How does the map $$Pic^0(i): Pic^0_{J/k} \rightarrow J$$ compare with the canonical principal polarization $$p \colon J \rightarrow Pic^0_{J/k}$$ induced by the $\Theta$-divisor? Are $Pic^0(i)$ and $p$ inverse to each other?

I came across Lemma 6.9 in the following notes http://www.jmilne.org/math/xnotes/JVs.pdf by James Milne, and, if I understand correctly, this lemma claims that $Pic^0(i) = -p^{-1}$. I am very confused about the minus sign though: if $X = J$ is an elliptic curve and $P = 0$ is the origin, aren't both $Pic^0(i)$ and $p$ ``the identity''?