If $X$ is a proper scheme over $k$, then the composition of $X\rightarrow \text{Spec}k\rightarrow \mathbb{P}_k^1$ is proper?

Question

If $X$ is a proper scheme over $k$, then the composition of $X\rightarrow \text{Spec}k\rightarrow \mathbb{P}_k^1$ is proper?

I thought of this question when I saw the answer of this question, if we take the sequence $X\rightarrow \text{Spec}k\rightarrow \mathbb{P}_k^1$, then I can't see how the Property $\mathscr{P}$ is used, because the Property $\mathscr{P}$ doesn't treat the property of the composition of two morphisms.

Thanks!

Answer 1

If $X$ is proper, then every morphism $f:X\rightarrow Y$ with $Y$ separated is proper. Indeed, consider a morphism $g:Y\rightarrow \mathrm{Spec}\,k$. Then $g\cdot f$ is proper (by properness of $X$) and $g$ is separated. Thus by cancellation (property $\mathcal{P}$), we deduce that $f$ is proper.