= Solution
A <morphism of schemes> is <proper morphism>[proper] when it is <finite type morphism>[of finite type], <separated morphism>[separated], and <universally closed morphism>[universally closed].
Each property is local on the target. Finite type is local by its affine definition. The restrictions of the <diagonal morphism> of $\varphi$ over the open sets $U_i$ are closed immersions; since being a closed subset is local on an open cover, the diagonal itself is a closed immersion, so $\varphi$ is separated. Finally, after any <base change of a morphism of schemes>[base change] $Y'\to Y$, the inverse images $U_i'=Y'\times_YU_i$ cover $Y'$. For every closed $C\subseteq X\times_YY'$, its image has closed intersection with every $U_i'$ because the restricted base-changed morphism is closed. The image is therefore closed in $Y'$. Thus $\varphi$ is universally closed and hence proper. This proves that <properness is local on the target>.
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