I was taking a look at a result in Tamás Szamuely's Galois groups and fundamental groups. The following argument can be found right after the proof of lemma 6.6.7.
Let $(K,\partial)$ be a differential field, let $(V,\nabla)$ be a differential module over $K$ and let $L$ be a Picard-Vessiot extension for the differential module $(V,\nabla)$. Szamuely claims that if $\sigma:L\rightarrow L$ is a $K$-automorphism of $L$ commuting with the derivation $\partial$, then the induced automorphism $\tilde{\sigma}:V\otimes_K L\rightarrow V\otimes_K L$ preserves the space of horizontal vectors $(V\otimes_K L)^{\nabla_L}$, and also that this is true for any $(W,\nabla)\in Ob(\langle V,\nabla\rangle_\otimes)$, where $\langle(V,\nabla)\rangle_\otimes$ is the full subcategory of the category of differential modules over $K$, $\mathbf{DiffMod}_K$ tensor-generated by $(V,\nabla)$.
The first problem I am having is that I do not see why this implies that the fibre functor $\omega_L:(W,\nabla)\mapsto (W\otimes_K L)^{\nabla_L}$ takes values not only in $\mathbf{Vecf}_k$, but in fact in the category of finite dimensional representations of $\mathrm{Gal}_\partial(L|K)(k)$. Why is it?
On the other hand, I do not see why is it that the functor that sends each $k$-algebra $R$ to $(W\otimes_K L\otimes_L R)^{\nabla_{L\otimes_K R}}$ allows us to deduce that $\omega_L$ induces a functor from $\langle V,\nabla\rangle_\otimes$ to the category of representations of the affine group scheme $\mathrm{Gal}_\partial(L|K)$, $\mathbf{Rep}(\mathrm{Gal}_\partial(L|K))$. Any idea why does this happen?
For the first question, I believe that since $\mathrm{Gal}_\partial(L|K)(k)$ is the set of $K$-automorphisms of $L$ that commute with $\partial$, we can send each $(W,\nabla)$ to an element of $\mathrm{GL}_W(k)\simeq\mathrm{Aut}(W)$, but I am not completely sure of this.