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

- 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.
- 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.