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Note: The terms "monic,epic,section,retraction" used in the post "Category Theory For Beginners - 2" and the terms "monomorphism,epimorphism,split monomorphism,split epimorphism" used in this one are respectively equivalent.



Let \( \mathcal{C} \) be a category. Show that

1. An isomorphism is a monomorphism and an epimorphism. Show that the inverse is not true in arbitrary categories. However, show that a morphism is an isomorphism if and only if it is a split monomorphism and a split epimorphism.
2. Let \( \displaystyle f : X \longrightarrow Y \) be a morphism in \( \mathcal{C} \). If \( \displaystyle \left( K, \phi \right) \) is the kernel of \( \displaystyle f \) and if \( \displaystyle \left( C, \psi \right) \) is the cokernel of \( \displaystyle f \) (assuming that both exist in \( \mathcal{C} \), show that \( \displaystyle \phi \) is a monomorphism and that \( \displaystyle \psi \) is an epimorphism.