Properness is local on the target

If $f:X\to Y$ is a morphism of schemes and $(U_i)$ is an open cover of $Y$, then $f$ is proper exactly when every restriction $f^{-1}(U_i)\to U_i$ is proper. The three defining properties—finite type, separatedness, and universal closedness—can each be checked on an open cover of the target.