In a seeded high-gain Free-electron Laser (FEL), where a coherent laser pulse interacts with an ultra-relativistic electron beam, the seed laser pulse can be frequency chirped, and the electron beam can be energy chirped. Besides these two chirps, the FEL interaction introduces an intrinsic frequency chirp in the FEL even if the above mentioned two chirps are absent. In this paper we examine the interplay of these three chirps. The problem is formulated as an initial value problem, and solved via a Green function approach. Besides the chirp evolution, we also give analytical expressions for the pulse duration and bandwidth of the FEL, which remains fully longitudinally coherent in the high gain exponential growth regime. Because the chirps are normally introduced for a final compression of the FEL pulse, some conceptual issues are discussed. We show that in order to get a short pulse duration, an energy chirp in the electron beam is necessary.