  The theory of MV cycles associated to a complex reductive group $G$ has
proven to be a rich source of structures related to representation theory. We
investigate double MV cycles, which are analogues of MV cycles in the case of
an affine Kac-Moody group. We prove an explicit formula for the
Braverman-Finkelberg-Gaitsgory crystal structure on double MV cycles,
generalizing a finite-dimensional result of Baumann and Gaussent. As an
application, we give a geometric construction of the Naito-Sagaki-Saito crystal
via the action of $\hat{SL}_n$ on Fermionic Fock space. In particular, this
construction gives rise to an isomorphism of crystals between the set of double
MV cycles and the Naito-Sagaki-Saito crystal. As a result, we can independently
prove that the Naito-Sagaki-Saito crystal is the $B(\infty)$ crystal. In
particular, our geometric proof works in the previously unknown case of
$\hat{\mathfrak{sl}}_2$.
