jats:titleAbstract</jats:title>jats:pPrevious works in this series have shown that an instance of a jats:inline-formulajats:alternativesjats:tex-math$$\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>-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 jats:italicjust</jats:italic> 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 Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the jats:inline-formulajats:alternativesjats:tex-math$$\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, jats:italicand</jats:italic> 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 jats:inline-formulajats:alternativesjats:tex-math$$\sqrt{8/3}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msqrt mml:mrow mml:mn8</mml:mn> mml:mo/</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula>-LQG surfaces with other topologies.</jats:p>