# disjoint union of projective morphisms is projective?

A morphism $f:X \to Y$ of schemes is projective if it factors into a closed immersion $i:X \to \mathbb{P}_{Y}^n(=\mathbb{P}_{\mathbb{Z}}^n\times_{Spec\mathbb{Z}} Y)$ for some $n$,followed by the projection $\mathbb{P}_{Y}^n \to Y$.

Let $f_{i}:X_{i} \to Y (i=1,2)$ are projective morphism of schemes. Is it true that $f_{1}\coprod f_{2}:X_{1}\coprod X_{2} \to Y$ is projective?

Yes. It suffices to show that $\mathbb{P}^m\coprod\mathbb{P}^n$ can be embedded in to some $\mathbb{P}^r$. Take $r\geq m+n+1$ and include the first copy as the coordinate subspace on the first $m+1$ coordinates and then second copy as the coordinate subspace on the last $n+1$ coordinates.
Writing $f_1\coprod f_2$ as $X_1\coprod X_2 \to \mathbb{P}_Y^{n_1}\coprod \mathbb{P}_Y^{n_2} \to \mathbb{P}^{N}_Y \to Y$ we see the desired result.