I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I form a new polyhedron, $B$, by gluing a cube with side-length $l$ to the base of $A$ ($B$ now has a base-length $l$ and height $2l$). Additionally, suppose that I have a new pyramid $A^{'}$ that is similar to $A$, but with a height and base-length of $2l$.

Does anyone know how to derive a conformal map from the interior of $B$ to that of $A^{'}$?