Remark on common properties of the products $ac$ and $ba$

In this paper, we discuss the common properties for the products $ac$ and $ba$ in various categories under the condition $a(ba)^{2}=abaca=acaba=(ac)^{2}a$. We prove that generalized Jacobson's lemma and Cline's formula are suitable for generalized n-strong Drazin invertible in rings, and generalized Jacobson's lemma is suitable for left and right Fredholm operator on Banach spaces.