Let \((X_{n})_{n \in \mathbb{N}}\) be a sequence of finite graphs with corresponding standard graph metrics \((d_{n})_{n \in \mathbb{N}}\) and let \((X, d)\) be the coarse disjoint union of \(\{ (X_{n}, d_{n})\}_{n \in \mathbb{N}}\). There exists a set of cut metrics \((d_{n,w})_{n \in \mathbb{N}}\) on \((X_{n})_{n \in \mathbb{N}}\) such that the identity function from \((X,d)\) to the coarse disjoint union \((X, d_{w})\) of \(\{(X_{n}, d_{n,w})\}_{n \in \mathbb{N}}\) is a coarse embedding if and only if there exists a coarse disjoint union, \((B,d_{B})\), of a sequence of hypercubes, \((B_{n})_{n \in \mathbb{N}}\), each equipped with the standard graph metric, \((d_{n,B})_{n \in \mathbb{N}}\), and a coarse embedding function \(f:(X, d) \to (B,d_{B})\) such that \(f(X_{n}) \subseteq B_{n}\).