Intelligent reflecting surface (IRS) has arisen as a promising technology to reconfigure the wireless propagation environment cost-effectively. Most of the existing works on IRS focused on the passive beamforming (PB) optimization and performance enhancement without considering the multiple inter-IRS links cooperation that did not reveal the full preponderance of the multi-IRS-assisted reconfigurable communication system. In this work, we investigate a double-IRS-assisted multiple input single output (MISO) downlink communication system with the active beamforming (AB) and the cooperative PB design in the absence of direct link. Taking both the double-reflection links and the single-reflection links into account, the AB at the base station (BS) and the cooperative PB at two IRSs are jointly optimized to maximize the weight sum rate (WSR) under the constraint of the transmit power. To tackle the problem, we propose the double-IRS-assisted fractional programming block coordinate descent (D-FPBCD) method to find the sub-optimal solution with low complexity. We first reconstruct the original issue as a tractable one by the closed-form fractional programming (FP) approach, then, the prox-linear block coordinate descent (BCD) and successive convex approximation (SCA) techniques are used to find the sub-optimal solution. Finally, simulation results demonstrate the effectiveness of the proposed double-IRS-assisted wireless communication scheme.