  In 1990, Schnyder used a 3-spanning-tree decomposition of a simple
triangulation, now known as the Schnyder wood, to give a fundamental
grid-embedding algorithm for planar maps. In the framework of mating of trees,
a uniformly sampled Schnyder-wood-decorated triangulation can produce a triple
of random walks. We show that these three walks converge in the scaling limit
to three Brownian motions produced in the mating-of-trees framework by
Liouville quantum gravity (LQG) with parameter $1$, decorated with a triple of
SLE$_{16}$'s curves. These three SLE$_{16}$'s curves are coupled such that the
angle difference between them is $2\pi/3$ in imaginary geometry. Our
convergence result provides a description of the continuum limit of Schnyder's
embedding algorithm via LQG and SLE.
