Continuous and uniformly continuous maps of powers of metric spaces

For every natural number n > 1, we construct a metric space X such that every continuous map ƒ: Xk → Xm is uniformly continuous whenever k ⩽ n − 1 but the category of all continuous maps of the spaces X, X2,…, Xn admits no full embedding into the category of all uniformly continuous maps of all powers of X and vice versa.