result in Sec. \[Sec:single\_to\_single\].

                  $w$ $p_c(w)$
  ------------------- ------------------
    $0              $ $0.4518     (1)$
    $10^{-4}        $ $0.4443     (2)$
    $3\times 10^{-4}$ $0.4405     (2)$
    $10^{-3}        $ $0.4353     (1)$
    $3\times 10^{-3}$ $0.4276     (1)$

  : \[Table:DI\_pc\] Critical points of the DI crossover model for some values of $w$’s (see the text). The numbers in the parentheses indicate the error of the last digits.

Since our argument for the crossover exponent is generally applicable to any universality class, we can check its validity through studying the similar type crossover between the directed Ising (DI) class models [@DIPark]. Consider a one-dimensional system with two species, say $A$ and $B$. Between the same species, hard core exclusion is applied, but different species can reside at the same site. The dynamic rules are as follows: The dynamics always starts with an $A$ particle. A randomly chosen $A$ particle can hop to one of nearest neighbors with probability $p$. If two $A$ particles meet at the same site by hopping, both particles are removed with probability $\frac{1}{2}$. If this annihilation attempt fails, the particle goes back to the original site. With probability $1-p$, a $B$ particle is generated at the same site occupied by the chosen $A$ particle. If that site 