= Solution
The scale is precisely the inverse <conformal map>:
$$
\sigma(w)=\frac{w}{1-w}=\phi^{-1}(w),\qquad
\sigma(0)=0,\quad\sigma(1)=\infty.
$$
By definition, a <chordal SLE in a specified scale> in $(\mathbb H,0,1)$ is a curve whose image under $\sigma$ is the half-plane-capacity-parameterized <SLE> from zero to infinity. Here
$$
\boxed{\sigma(\widetilde\gamma_t)
=\sigma(\phi(\gamma_t))=\gamma_t,}
$$
and the right-hand side is the given $\operatorname{SLE}_6$. Equivalently, this is <Conformal invariance of SLE>, with the target point transported from infinity to $1$. Thus \b[$\widetilde\gamma$ is $\operatorname{SLE}_6$ in $(\mathbb H,0,1)$ in scale $\sigma$], using the original parameter $t$. This scaled parameter is distinguished from the ordinary <half-plane-capacity parameterization> of its image hulls used next.
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