# how to define the composition of two dominant rational maps?

Let's follow Hartshorne's definition, according to which, a rational map $\phi:X \rightarrow Y$ from variety $X$ to variety $Y$ is an equivalence class $\langle U,\phi_U \rangle$, where $U$ is open in $X$ and $\phi_U: U \rightarrow Y$ is a morphism of varieties. If $\phi_U(U)$ is dense in $Y$, $\phi$ is called dominant.

Hartshorne says that we can "clearly compose dominant rational maps". So let $\phi: X \rightarrow Y, \psi:Y \rightarrow Z$ be dominant rational maps given by the pairs $\langle U,\phi_U \rangle, \langle V, \psi_V \rangle$. Question: how exactly do we define $\psi \circ \phi$? To be consistent with the definition, we need to construct a pair $\langle W,\sigma_W \rangle$, where $W$ is open in $X$ and $\sigma_W : W \rightarrow Z$ is a morphism of varieties with dense image and somehow this $\sigma_W$ must arise from the "composition" of $\phi_U$ and $\psi_V$. But how can we compose these two?

Remark: i am aware that there are at least two other questions on this forum related to this topic, however the answers and comments do not clarify the issue i am raising.

Let $W$ be the intersection of $U$ (a domain for $\phi$) with $\phi^{-1}(V)$, where $V$ is a domain for $\psi$. This is the intersection of two non-empty open subsets of $X$ (we use dominance of $\phi$ to deduce that $\phi^{-1}(V)$ is non-empty). We then let $\sigma_W$ be the composite of $\phi_W$ (this is just the restriction of $\phi_U$ to $W$) and $\psi_V$.