Why is this monic equalizer an isomorphism?

Question

What in this proof shows that the sentence in the red box below (from Mac Lane's category theory) is an isomorphism?

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Answer 1

From $e_1se=1_v$ follows that $e_1$ is a split epimorphism. On the other hand $e_1$ is monic, because equalizers are monic. Thus $e_1$ is a monic split epimorphism, hence an isomorphism.

More precisely, from $e_1se=1_v$ we get $e_1see_1=e_1$ and since $e_1$ is monic, it's left-cancellable thus giving $see_1=1$. This proves $e_1^{-1}=se$.