  Previous works in this series have shown that an instance of a
$\sqrt{8/3}$-Liouville quantum gravity (LQG) sphere has a well-defined distance
function, and that the resulting metric measure space (mm-space) agrees in law
with the Brownian map (TBM). In this work, we show that given just the mm-space
structure, one can a.s. recover the LQG sphere. This implies that there is a
canonical way to parameterize an instance of TBM by the Euclidean sphere (up to
M\"obius transformation). In other words, an instance of TBM has a canonical
conformal structure.
  The conclusion is that TBM and the $\sqrt{8/3}$-LQG sphere are equivalent.
They ultimately encode the same structure (a topological sphere with a measure,
a metric, and a conformal structure) and have the same law. From this point of
view, the fact that the conformal structure a.s. determines the metric and
vice-versa can be understood as a property of this unified law. The results of
this work also imply that the analogous facts hold for Brownian and
$\sqrt{8/3}$-LQG surfaces with other topologies.
