The core and dual core inverses of morphisms with kernels

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