This work is concerned with the output regulation problem for a non-autonomous infinite dimensional system. Specially, a regulator for boundary controlled time-varying hyperbolic systems is designed. The disturbances can act within the space domain, and affect both boundaries and the to-be-controlled output. The to-be-controlled output comprises in-domain pointwise, distributed and boundary outputs. The output regulation problem is solved in finite time. The regulator design is based on the solvability of the regulator equations. Due to the time-varying setting of the system and the generality of the to-be-controlled output, solving regulator equations becomes more challenging compared to the case of autonomous systems. A novel method is introduced to overcome this difficulty. By considering the regulator equations as a control system, we examine the dual system of the regulator equations and transform the solvability of the regulator equations into the validity of an observability-like inequality. Under the conditions regarding the boundary coupling term and the to-be-controlled output, we have proven this inequality. Additionally, time-varying setting also brings an advantage to the problem. Since the regulator equations is time-dependent, its solvability does not depend on the eigenmodes of the signal model. On the contrary, in the case of autonomous systems, its solvability depends on the relationship between the plant transfer behavior and the eigenmodes of the signal model.