I'm working on Moonen's notes, Charpter 2. Corollary 2.10 shows that for fixed line bundle $L$ on Abelian variety $X$, $\varphi_L:X\rightarrow \mathrm{Pic}(X), x\mapsto [t_x^*L\otimes L^{-1}]$ is a homomorphism where $t_x:a\mapsto a+x$ is the translation.
Now we calculate this map explicitly for elliptic curve $E$ with fixed point $O$. I have make it clear that $t_x^* \mathcal{O}_E(D)=\mathcal{O}_E(D-x)$ for any divisor $D$, but get stuck when plug in $L=\mathcal{O}_E$ to $\varphi_L$. For $L=\mathcal{O}_E(O)$, I calculate as following: $$t_x^*L\otimes L^{-1}=\mathcal{O}_E(O-x)\otimes \mathcal{O}_E(-O)=\mathcal{O}_E(-x)$$ but what I want is the result $\mathcal{O}_E(O-x)$. Where is the problem?