Complex conjugate and tensor product Let $V$ be a real vector space and $f : V \rightarrow V$ a linear endomorphism. Also, let $\sigma : \mathbb{C} \rightarrow \mathbb{C}$ be complex conjugation.

If $A$ is a real matrix, then it is clear how to interpret $A$ as a complex matrix and what the the claim $\sigma(A) = A$ means. I am confused how to write this for the abstract $f$.

The complex interpretation of $f$ is $f \otimes \mathrm{id} : V \otimes_{\mathbb{R}} \mathbb{C} \rightarrow V \otimes_{\mathbb{R}} \mathbb{C}.$ I would like to understand what the complex interpretation of "$\sigma(f) = f$" is. My thought was that applying $\sigma$ to $f \otimes \mathrm{id}$ should give us $f \otimes \sigma$, but I don't think this is true after looking at $f = \mathrm{id}_V$ (for example).

Sorry if my question is unclear.
The maps $\mathbb{C}\rightarrow\mathbb{C}$ are just multiplication by complex numbers: $z\overset{w}{\mapsto}wz$. Then $\sigma$ sends $f\otimes w$ to $f\otimes \sigma(w)$.
The maps $\mathbb{C}\rightarrow\mathbb{C}$ are just multiplication by complex numbers: $z\overset{w}{\mapsto}wz$. Then $\sigma$ sends $f\otimes w$ to $f\otimes \sigma(w)$.