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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 $\\kappa\\lambda$ and $\\varphi^{\\ast}\\varphi^3+\\kappa^{\\ast}\\kappa$ are invertible. In this case, we give the representation of the core inverse of $\\varphi$. 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