We study uniform continuity for functions on coinductive data. The familiar 1-Lipschitz case yields the standard notion of causality from synchronous systems. We generalise this to fan-causality, where the output observed at depth n may depend on input available at a depth specified in terms of n by a monotone, unbounded map on natural numbers. This captures computations whose look-ahead varies with observation depth, including bounded fixed look-ahead as a limiting case. Final coalgebras constructed as a limit of finite iterations of a functor carry a canonical metric. We show that fan-causal maps on such final coalgebras correspond precisely to uniformly continuous maps with respect to this metric. We then construct a symmetric monoidal category whose morphisms are in bijection with fan-causal maps. This yields a compositional foundation for convenient specifications and reasoning tools for general uniformly continuous maps on final coalgebras.