Let R be an arbitrary ring with identity and M be a right R-module with S = End(M-R). Let f is an element of S. f is called pi-morphic if M/f(n)(M) congruent to r(M)(f(n)) for some positive integer n. A module M is called pi-morphic if every f is an element of S is pi-morphic. It is proved that M is pi-morphic and image-projective if and only if S is right pi-morphic and M generates its kernel. S is unit-it-regular if and only if M is pi-morphic and pi-Rickart if and only if M is pi-morphic and dual pi-Rickart. M is pi-morphic and image-injective if and only if S is left pi-morphic and M cogenerates its cokernel.