  We present a proof of the compositional shuffle conjecture, which generalizes
the famous shuffle conjecture for the character of the diagonal coinvariant
algebra. We first formulate the combinatorial side of the conjecture in terms
of certain operators on a graded vector space $V_*$ whose degree zero part is
the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend
these operators to an action of an algebra $\tilde{\AA}$ acting on this space,
and interpret the right generalization of the $\nabla$ using an involution of
the algebra which is antilinear with respect to the conjugation $(q,t)\mapsto
(q^{-1},t^{-1})$.
