I am being stuck by the proof of the existence of Poincare line bundle of complex torus in Griffiths-Harris. Here is the question:

Let $M$ be a complex torus and $M'$ be the complex torus dual to $M$ (the one consists of all holomorphic line bundle whose chern class is zero). By Kunneth's formula and the definition of $M',$ one knows that the identity element $I$ of $$ Hom(H^1(M,\mathbb{Z}),H^1(M,\mathbb{Z}))\equiv H^1(M,\mathbb{Z})\otimes H^1(M',\mathbb{Z})$$ belongs to $H^2(M\times M',\mathbb{Z})$ and is of type (1,1). Hence, there exists a line bundle $P$ over $M\times M'$ having $I$ as its chern class. For each $\xi$ in $M'$, let $P_{\xi}$ be the restriction of $P$ on $M\times {\xi}$, clearly it is in $M'$. Hence we get a mapping $\Phi: M' \rightarrow M'$ that send $\xi$ to $P_{\xi}.$

Why the induced mapping of $\Phi$ on the first homology is the identity?

Thank you so much.