Here a proof of Schur's lemma for representations is given. The proof begins by letting $f:V\to V'$ be a 'homomorphism of $G$-modules', but clicking on that link, I don't see how you can conclude that $f$ is a homomorphism. In fact, multiplication on $V$ and $W$ do not seem to be defined there.
2026-04-07 23:45:31.1775605531
Why is $f$ a homomorphism in this proof?
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You can conclude that $f$ is a homomorphism (of vectorspaces) as the definition of a homomorphism of $G$-modules includes the assumption that $f$ is a linear map a.k.a. a homomorphism (of vertorspaces).