Analyticity of the outer function of an analytic composition

Question

Let $\mathscr{U}$ be an open neighborhood of the origin of $\mathbb{C}$ and let $F(t,x)$ be a function that is continuous on $\mathbb{C} \times \mathscr{U}$ and that is holomorphic in $\mathscr{U}$ for each fixed $t \in \mathbb{C}$.

Moreover, let $T > 0$ and let $u(t,x)$ be a holomorphic function on $B_T \times \Omega$ where $B_T = \{ t \in \mathbb{C} : |t| < T \}$ and $\Omega$ is an open subset of $\mathbb{C}$.

Suppose:

1. $(t,x) \in B_T \times \Omega \Rightarrow u(t,x) \in \mathscr{U}$;
2. $F(t,u(t,x))$ is holomorphic on $B_T \times \Omega$.

Question: Is $F(t,z)$ also holomorphic in $t$?