  In this paper, we introduce two (anti)homomorphisms that preserve the Haar
state values of monomials. Together with the modular automorphism, the three
(anti)homomorphisms are used in our new algorithm to compute the Haar states of
monomials on $\mathcal{O}(SL_q(3))$. Comparing with the algorithm proposed in
the author's previous work \cite{lu2023}, the new algorithm reduces the linear
relations used in the computation to a half.
