Projective varieties: rational maps and morphisms of schemes

Question

I'm a beginner with algebraic varieties topics, and I studied a (very little) of scheme before. The fact is that I don't truly manage to correctly understand the bridge between the different notions of "morphism" I encountered. Actually, let $X$, $Y$ be a projective varieties, $X=\operatorname{Proj} (k[T_0, \dots, T_n]/I)$, where $I$ is an homogenous ideal and $k$ a field (algebraically closed if needed), and $Y=\operatorname{Proj}(k[T_0, \dots, T_n]/J)$, where $J$ is an homogenous ideal. There are two notions of morphisms $Y \to X$ I saw :

1. $X$ and $Y$ are schemes, and then we have the usual meaning of morphisms of schemes, and so a morphism $Y \to X$ is a $k$-morphism in the sense of schemes;
2. The point of view of rational map, so a morphism $Y \to X$ is an equivalence class of tuples $\phi=(\phi_0: \dots : \phi_n)$, $\phi_i \in \overline{k}(X)$, defined at $P$ for all $P \in Y$, and not all zero.

My question is, how to show that those two definitions coincide ?

Thank you!