Take $E$ a $K$-vector space with $K$ a field with characteristic $0$. Then take $$ F:\operatorname{End}_K(E) \to K $$ a map such that $$F(1)=1$$ and $$ F(\phi\circ\psi) = F(\phi)F(\psi) $$ Then there exists a $f:K \to K$ such that is multiplicative and $F(\phi) = f(\det\phi)$ for any $\phi \in \operatorname{End}_K(E)$.
This is an exercise by Linear Algebra of Werner Greub and the hint is
Can anyone give me a way to go from the hint to the answer? I have thought to write any endomorphism using the maps he introduces but the expression I find is not so clear in terms of determinant