Induced map from the universal cover of the base space to the Teichmuller space of the fiber is holomorphic

Question

Let $\phi:X\to Y$ be a submersion and holomorphic map with not every two fibers are biholomorphic, where $X$ a compact complex surface, $Y$ is a Riemann surface. Let $\pi:U\to Y$ be the universal covering space of $Y$ and $T$ be a Teichmuller space of Teichmuller structures on a compact topological surface of genus same as the genus of the fiber of $p$. Then we get a map $f:U\to T$, s.t. $u\in U$ is mapped to a point of $T$ representing the complex structure of $\phi^{-1}(\pi(u))$. Then why is $f$ holomorphic?