The core and dual core inverse of a morphism with factorization

Let $\mathcal{C}$ be a category with an involution $\ast$. Suppose that $\phi :X\to X$ is a morphism and $({\phi }_1,Z,{\phi }_2)$ is an (epic, monic) factorization of $\phi$ through $Z$, then $\phi$ is core invertible if and only if $({\phi }^{\ast }{)}^2{\phi }_1$ and ${\phi }_2{\phi }_1$ are both left invertible if and only if $(({\phi }^{\ast }{)}^2{\phi }_1,Z,{\phi }_2)$, $(\varphi_2^{\ast}, Z, \varphi_1^{\ast}\varphi^{\ast}\varphi)$and$(\varphi^{\ast}\varphi_2^{\ast}, Z, \varphi_1^{\ast}\varphi)$ are all essentially unique (epic, monic) factorizations of $({\phi }^{\ast }{)}^2\phi$ 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 $R$.