Definition of Functor in Waterhouse

Question

On page 21 of Waterhouse's Introduction to Affine Group Schemes he defines (on objects, which are $k$-algebras $R$) a functor $GL_V(R) = \mathrm{Aut}_R(V\otimes R)$ where $V$ is a fixed $k$-module ($k$ a field), which goes from the category of $k$-algebras to the category of groups. He does not define what this functor does to morphisms, so I was wondering what it does.

Answer 1

Given a $k$-algebra homomorphism $f:R\to S$, we can consider $S$ as an $R$-algebra via $f$, and there is then a canonical isomorphism $V\otimes_k S\cong (V\otimes_k R)\otimes_R S$. This lets us turn an automorphism of $V\otimes_kR$ into an automorphism of $V\otimes_k S$, since $-\otimes_R S$ is a functor.

Or, to see things very concretely, pick a basis for $V$, say with $n$ elements. Then $GL_V(R)$ can be identified with the group of invertible $n\times n$ matrices with entries in $R$. Given a homomorphism $f:R\to S$, we then get a homomorphism $GL_V(R)\to GL_V(S)$ by just applying $f$ to each matrix entry.