Choose $o^t\in X^t$ so that $t\mapsto o^t$ is smooth and yet choose a famity of unit vectors $u^t\in T_{o^t}X^t$, so that $t\mapsto u^t$ is smooth.

Parametrize $X^t$ isothermaly by unit disc $f^t:D\to X^t$ in such a way that $f^t(0)=o^t$ and $(d_0f^t)(1)$ is proportional to $u^t$. Such a parametrization is unique. [The later follows since conformal diffeomorphism $h:D\to D$ such that $h(0)=0$ and $h'(0)\in\mathbb R_+$ has to be identity.]

It follows that $t\mapsto f^t$ is continuous; otherwise different partial limits would give different isothermal coordinate with chousen origin and real direction.

Choose $o^t\in X^t$ so that $t\mapsto o^t$ is smooth and yet choose a famity of unit vectors $u^t\in T_{o^t}X^t$, so that $t\mapsto u^t$ is smooth.

Parametrize $X^t$ isothermaly by unit disc $f^t:D\to X^t$ in such a way that $f^t(0)=o^t$ and $(d_0f^t)(1)$ is proportional to $u^t$. Such a parametrization is unique. [The later follows since conformal diffeomorphism $h:D\to D$ such that $h(0)=0$ and $h'(0)\in\mathbb R_+$ has to be identity.]

It follows that $t\mapsto f^t$ is continuous; otherwise different partial limits would give different isothermal coordinate with chousen origin and real direction.

With a bit more work, one can show that $t\mapsto f^t$ is smooth.

Choose $o^t\in X^t$ so that $t\mapsto o^t$ is smooth and yet choose a famity of unit vectors $u^t\in T_{o^t}X^t$, so that $t\mapsto u^t$ is smooth.

Parametrize $X^t$ isothermaly by unit disc $f^t:D\to X^t$ in such a way that $f^t(0)=o^t$ and $(d_0f^t)(1)$ is proportional to $u^t$. Such a parametrization is unique. It follows that $t\mapsto f^t$ is continuous; otherwise different partial limits would give different isothermal coordinate with chousen origin and real direction.

Choose $o^t\in X^t$ so that $t\mapsto o^t$ is smooth and yet choose a famity of unit vectors $u^t\in T_{o^t}X^t$, so that $t\mapsto u^t$ is smooth.

Parametrize $X^t$ isothermaly by unit disc $f^t:D\to X^t$ in such a way that $f^t(0)=o^t$ and $(d_0f^t)(1)$ is proportional to $u^t$. Such a parametrization is unique. [The later follows since conformal diffeomorphism $h:D\to D$ such that $h(0)=0$ and $h'(0)\in\mathbb R_+$ has to be identity.]

It follows that $t\mapsto f^t$ is continuous; otherwise different partial limits would give different isothermal coordinate with chousen origin and real direction.