Let $P_1=(1:0:0),P_2=(0:1:0),P_3=(0:0:1) \in \mathbb{P}_2$ (over an algebraic closed field). Denote $U=\mathbb{P}^2 \setminus \{P_1,P_2,P_3 \}$ and consider the map
$$ f:U \rightarrow \mathbb{P}^2, (a_0:a_1:a_2) \mapsto (a_1a_2:a_0a_2:a_0a_1) $$
Let $\tilde{\mathbb{P}}^2$ be the bow up at $\{P_1,P_2,P_3 \}$.
According to the Andreas Gathmann notes, there is an isomorphism $\tilde{f}:\tilde{\mathbb{P}}^2 \rightarrow \tilde{\mathbb{P}}^2$ that extends $f$.
Can someone explain me, or give me a hint on how can we proove that?