Monte Carlo simulation on the crystallization of double crystalline diblock copolymer unravels an intrinsic relationship between block asymmetry and crystallization behaviour.We model crystalline A-B diblock copolymer, wherein the melting temperature of A-block is higher than that of the B-block. We explore the composition dependent crystallization behaviour by varying the relative block length with weak and strong segregation strength between the blocks. In weak segregation limit, we observe that with increasing the composition of B-block, its crystallization temperature increases accompanying with higher crystallinity. In contrast, A-block crystallizes at a relatively low temperature along with the formation of thicker and larger crystallites with the increase in B-block composition. We attribute this non-intuitive crystallization trend to the dilution effect imposed by B-block.When the composition of the B-block is high enough, it acts like a "solvent" during the crystallization of A-block. A-block segments are more mobile and hence less facile to crystallize, resulting depression in crystallization temperature with the formation of thicker crystals. At strong segregation limit, crystallization and morphological development are governed by the confinement effect, rather than block asymmetry. Isothermal crystallization reveals that the crystallization follows a homogeneous nucleation mechanism with the formation of two dimensional crystals. Two-step, compared to one-step isothermal crystallization leads to the formation of thicker crystals of A-block due to the dilution effect of the B-block.