The core and dual core inverses of morphisms with kernels

Let $mathscr{C}$ be an additive category with an involution $ast$. Suppose that $varphi : X rightarrow X$ is a morphism with kernel $kappa : K rightarrow X$ in $mathscr{C}$, then $varphi$ is core invertible if and only if $varphi$ has a cokernel $lambda: X rightarrow L$ and both $kappalambda$ and $varphi^{ast}varphi^3+kappa^{ast}kappa$ are invertible. In this case, we give the representation of the core inverse of $varphi$. We also give the corresponding result about dual core inverse.