The composition of conformal mappings is conformal.
Suppose $F:U\rar V$ and $G:V\rar W$ are conformal, then for all $x\in U$, all $y\in V$ and all $u,v\in\R^n$:
$$
\la DF(x)^*DF(x)u,u\ra=f(x)^2\Vert u\Vert^2
\quad\mbox{and}\quad
\la DG(y)^*DG(y)v,v\ra=g(y)^2\Vert v\Vert^2~.
$$
By the chain rule we have $D(G\circ F)(x)=DG(F(x))DF(x)$. Putting $y\colon=F(x)$ and $v\colon=DG(y)u$ we thus get:
\begin{eqnarray*}
\la D(G\circ F)(x)^*D(G\circ F)(x)u,u\ra
&=&\la DG(y)^*DF(x)^*DF(x)DG(y)u,u\ra\\
&=&\la DF(x)^*DF(x)v,v\ra
=f(x)^2\la v,v\ra\\
&=&f(x)^2\la DG(y)u,DG(y)u\ra
=f(x)^2g(y)^2\Vert u\Vert^2~.
\end{eqnarray*}
Proving that $G\circ F$ is conformal with scaling function $h(x)\colon=f(x)g(F(x))$.