In this paper, we first study agreement protocols for coupled continuous-time nonlinear dynamics over cooperative multi-agent networks. To guarantee convergence for such systems, it is common in the literature to assume that the vector field of each agent is pointing inside the convex hull formed by the states of the agent and its neighbors given the relative states between each agent and its neighbors are available. This convexity condition is relaxed in this paper, as we show that it is enough that the vector field belongs to a strict tangent cone based on a local supporting hyperrectangle. The new condition has the natural physical interpretation of adding a compass for each agent in addition to the available local relative states, as each agent needs only to know in which orthant each of its neighbor is. It is proven that the multi-agent system achieves exponential state agreement if and only if the time-varying interaction graph is uniformly jointly quasi-strongly connected. Cooperative-antagonistic multi-agent networks are also considered. For these systems, the (cooperative-antagonistic) relation has a negative sign for arcs corresponding to antagonistic interactions. State agreement may not be achieved for cooperative-antagonistic multi-agent systems. Instead it is shown that asymptotic modulus agreement is achieved if the time-varying interaction graph is uniformly jointly strongly connected.