A causal manifold $(M,\lambda)$ is a real manifold $M$ endowed with a closed convex proper cone $\lambda$ in its cotangent bundle $T^*M$. On such a manifold, one defines the $\lambda$-topology and the past or the future of any subset. A time function is a smooth surjective causal map $q: M\to\mathbb R$ proper on the past or future of any compact subset of $M$. Using a time function, we show that if the micro-support of a sheaf $F$ does not intersect $\lambda\cup-\lambda$ outside of the zero-section, then for any Cauchy hypersurface $N_t=q^{-1}(t)$, the restriction morphism $\mathrm{R}\Gamma(M;F)\to\mathrm{R}\Gamma(N_t;F\vert_{N_t})$ is an isomorphism. As an application, we get that the Cauchy problem is globally well-posed for hyperfunction solutions of hyperbolic systems.