Quiver varieties and adjoint crystals of level $\ell$ for type $A_n^{(1)}$

Let $B(\Lambda)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(\Lambda)$ of the crystal $B(\Lambda)$ using quiver varieties and the path realization $\mathcal{P}^{\rm ad}(\Lambda)$ arising from the adjoint crystal $\Bad{\ell}$ of level $\ell$.