  We study the graded limits of simple
$U_q(\tilde{\mathfrak{sl}}_{n+1})$-modules which are isomorphic to tensor
products of Kirillov-Reshetikhin modules associated to a fix fundamental
weight. We prove that every such module admits a graded limit which is
isomorphic to the fusion product of the graded limits of its tensor factors.
Moreover, using recent results of Naoi, we exhibit a set of defining relations
for these graded limits.
