Morphism of projective varieties defined locally

Question

Let $X$ be a projective variety and $(U_i)_{i\in I}$ be a covering of $X$ of locally closed irriducible subsets (quasi-projective subvarieties of $X$). I'm trying to check that if $f\colon X\longrightarrow Y$ is a map of projective varieties such that $f_{|U_i}\colon U_i\longrightarrow Y$ is a morphism then $f$ is morphism. Any help?