Suppose $X$ and $Y$ are smooth projective $T$-varieties, where $T$ is a smooth affine curve. Let $\phi:X\dashrightarrow Y/T$ be a rational map over $T$. My question is: is there a closed subset $Z\subset X$ such that $codim_{X_t}(Z\cap X_t)\geq 2$ for all $t\in T$ such that $\phi_{\vert X\setminus Z}$ is actually a morphism of $T$-varieties?
I know that each restricted map $\phi_t:X_t\dashrightarrow Y_t$ extends in codimension one on $X_t$ for all $t\in T$. Is it obvious that these extension "fit together" over $T$?