Given two compact, connected manifolds with corners, and a navigation function (NF, a refined artificial potential function) on each manifold, this paper presents a simple composition law that yields a new NF on the cross product space. The method provides tunable "hooks" for shaping the new potential function while still guaranteeing obstacle avoidance and essentially global convergence. The composition law is associative, and successive compositions fold into a single, computational simple expression, enabling the practical construction of NFs on the Cartesian product of several manifolds.