Isomorphism Preserves Identity/Proof 2

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Theorem

Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.


Then $\circ$ has an identity $e_S$ if and only if $\map \phi {e_S}$ is the identity for $*$.


Proof

We have that an isomorphism is a homomorphism which is also a bijection.

By definition, an epimorphism is a homomorphism which is also a surjection.

That is, an isomorphism is an epimorphism which is also an injection.

Thus Epimorphism Preserves Identity can be applied.

$\blacksquare$