We report dynamic Monte Carlo simulations of microphase separated diblock copolymers, to investigate how crystallization of one species could accelerate the subsequent crystallization of another species. Although the lattice copolymer model brings a boundary constraint to the long periods of microdomains, the single-molecular-level force balance between two blocks and its change can be revealed in this simple approach. We found two contrastable acceleration mechanisms: (1) the metastable lamellar crystals of one species become thicker at higher crystallization temperatures, sacrificing its microphase interfacial area to make a larger coil-stretching of another amorphous species and hence to accelerate subsequent crystallization of the latter with a more favorable conformation. (2) While in the case allowing chain-sliding in the crystal, the equilibrated lamellar crystals of one species become thinner at higher temperatures, sacrificing its thermal stability to gain a higher conformational entropy of another amorphous species and hence to accelerate subsequent crystallization of the latter with a stronger tension at the block junction. Parallel situations of experiments have been discussed.