  In this paper, we prove one case of the conjecture given by Hernandez and
Leclerc\cite{HL0}. Specifically, we give a cluster algebra structure on the
Grothendieck ring of a full subcategory of the finite dimensional
representations of a simply-laced quantum affine algebra $U_q(\widehat{\g})$.
In the procedure, we also give a specific description of compatible subsets of
type $E_{6}$. As a conclusion, for every exchange relation of cluster algebra
there exists a exact sequence of the full subcategory corresponding to it.
