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Suppose that $\\varphi : X \\rightarrow X$ is a morphism of $\\mathscr{C}$ with core inverse $\\varphi^{\\co} : X \\rightarrow X$ and $\\eta : X \\rightarrow X$ is a morphism of $\\mathscr{C}$ such that $1_X+\\varphi^{\\co}\\eta$ is invertible. 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