Rational map of varieties

Definition
Let $$X$$ and $$Y$$ be varieties over a field $$k$$ (which we usually assume to be algebraically closed. A rational map $$\phi:X \to Y$$ is an equivalence class of:

Pairs $$$$ where $$U$$ is an open set in $$X$$ and $$\phi_U$$ is a morphism of varieties from $$U$$ to $$Y$$

under the equivalence relation:

$$$$ if and only if $$\phi_U$$ and $$\phi_V$$ agree on $$U \cap V$$.