morphism to projective space homogeneous coordinates

Let $X\subseteq {\mathbf{P}}_k^n$ be a classical projective variety. A map $\phi :X\to {\mathbf{P}}_k^m$ is a projective morphism if there is an open cover $X={\bigcup }_{\alpha }{U}_{\alpha }$ such that, for every $\alpha$, homogeneous polynomials $F_{\alpha ,0},\dots ,F_{\alpha ,m}$ of one common degree have no common zero on $U_{\alpha }$ and $\phi (p)=[F_{\alpha ,0}(p):\cdots :F_{\alpha ,m}(p)](p\in U_{\alpha })\mathrm{.}$ On each $U_{\alpha }\cap \{F_{\alpha ,i}\mathrm{\ne }0\}$ the target-chart coordinates are the regular functions $F_{\alpha ,j}\mathrm{/}{F}_{\alpha ,i}$. The local tuples define one map precisely when, for every $p\in U_{\alpha }\cap U_{\beta }$, $F_{\alpha ,i}(p)F_{\beta ,j}(p)=F_{\alpha ,j}(p)F_{\beta ,i}(p)\text{for all }i,j;$ equivalently, they give the same projective point there. Multiplying every entry of one tuple by a common locally nonvanishing regular factor changes no point.