The core and dual core inverse of a morphism with factorization.

Let 𝒞 be a category with an involution ∗. Suppose that φ : X → X is a morphism and (φ 1 , Z , φ 2) is an (epic, monic) factorization of φ through Z , then φ is core invertible if and only if (φ ∗) 2 φ 1 and φ 2 φ 1 are both left invertible if and only if ((φ ∗) 2 φ 1 , Z , φ 2) , (φ 2 ∗ , Z , φ 1 ∗ φ ∗ φ) and (φ ∗ φ 2 ∗ , Z , φ 1 ∗ φ) are all essentially unique (epic, monic) factorizations of (φ ∗) 2 φ through Z. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an R -morphism in the category of R -modules of a given ring R. [ABSTRACT FROM AUTHOR]