Let $\pi:X\to T$ be a proper surjective holomorphic mapping of maximal rank from a complex manifold $X$ to a complex manifold $T$. Let $X_t=\pi^{-1}(t)$. Then $X_t$ is the fibre over $t$, or the compact complex submanifold of $X$ corresponding to the parameter $t\in T$.
I can't see clearly why $\psi_p|_{U_p\cap X_t}\cong U'_p\cap C^n\times\{t\}$ by implicit function. I think by implicit function theorem we can only get the case for $t=0$. However I can't see we can find a general $U_p$ s.t. the $t$ around $0$ can share with $0$ by using the common $U_p$.