  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.
