In this paper, we report on the development and manufacture of a dual-slit electron-optical dissector based on the PIF-01 streak image tube with a picosecond time resolution operating in the crossed-sweep mode. The dissector is designed to work on synchrotron radiation sources and electron-positron colliders, when it is necessary to record simultaneously the temporal profiles of electron and positron bunches filling from two to several hundred neighboring separatrices of an accelerator ring. In order to obtain the crossed-sweep mode, high-frequency (HF) sinusoidal voltages with multiple frequencies are fed to two pairs of deflecting plates of the tube, located perpendicular to each other. This separation of pulses will allow separate acquisition of the bunch profiles from neighboring separatrices of an accelerator, which are superimposed on each other in the absence of the crossed sweep. During the tests of the device which have been conducted on a laser system, two trains of light pulses, "shifted" in time with respect to each other, were recorded with a time resolution of 6 ± 0.5 ps. Improvement of a single slit dissector was also done using the MASIM software. As a result, the temporal resolution of the modernized dissector measured on a femtosecond Ti:sapphire laser was about 1.5 ps. The dissector was tested on the MLS (Berlin, Germany) accelerator. Using both the HAMAMATSU C10910 streak camera and the dissector, comparative measurements of the longitudinal bunch length were performed.