  In a martingale optimal transport (MOT) problem mass distributed according to
the law $\mu$ is transported to the law $\nu$ in such a way that the martingale
property is respected. Beiglb\"ock and Juillet (On a problem of optimal
transport under marginal martingale constraints, Annals of Probability,
44(1):42-106, 2016) introduced a solution to the MOT problem which they
baptised the left-curtain coupling. The left-curtain coupling has been widely
studied and shown to have many applications, including to martingale
inequalities and the model-independent pricing of American options. Beiglb\"ock
and Juillet proved existence and uniqueness, proved optimality for a family of
cost functions, and proved that when $\mu$ is a continuous distribution, mass
at $x$ is mapped to one of at most two points, giving lower and upper
functions. Henry-Labord\`ere and Touzi (An explicit martingale version of
Brenier`s theorem, Finance and Stochastics, 20:635-668, 2016) showed that the
left-curtain coupling is optimal for an extended family of cost functions and
gave a construction of the upper and lower functions under an assumption that
$\mu$ and $\nu$ are continuous, together with further simplifying assumptions
of a technical nature. In this article we construct these upper and lower
functions in the general case of arbitrary centred measures in convex order,
and thereby give a complete construction of the left-curtain coupling. In the
case where $\mu$ has atoms these upper and lower functions are to be
interpreted in the sense of a lifted martingale.
