Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor.

Let $(\mathcal{S},w')$ be a full Tannakian sub-category (i.e. closed under tensor products, direct sums, duals and quotients) as well as taking subobjects/subquotients. Denote its fundamental group by $H$.

In "Deligne, Pierre, and J. S. Milne. "Tannakian categories." Lecture Notes in Matehmatics (2012)", Proposition 2.21, it is said that the natural morphism $G\longrightarrow H$ is faithfully flat (and in particular surjective). Their proof is tautological, and I could not understand the tautology.

The way the authors deduce the claim is by arguing that the induced map on the underlying algebras (taking the duals of $G = \text{Aut}^{\otimes}(w) \longrightarrow \text{Aut}^{\otimes}(w') = H$), the map

$$\text{Aut}^{\otimes}(w')^{\vee}\longrightarrow \text{Aut}^{\otimes}(w)^{\vee}$$ is clearly injective. While I understand how the conclusion follows from this implication, I would love for an explanation of why this is the case, or an alternative argument.

A reference would also be appreciated.

Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor.

Let $(\mathcal{S},w|_{\mathcal{S}})$ be a full Tannakian sub-category (i.e. closed under tensor products, direct sums, duals and quotients) as well as taking subobjects/subquotients. Denote its fundamental group by $H$.

In "Deligne, Pierre, and J. S. Milne. "Tannakian categories." Lecture Notes in Matehmatics (2012)", Proposition 2.21, it is said that the natural morphism $G\longrightarrow H$ is faithfully flat (and in particular surjective). Their proof is tautological, and I could not understand the tautology.

The way the authors deduce the claim is by arguing that the induced map on the underlying algebras (taking the duals of $G = \text{Aut}^{\otimes}(w) \longrightarrow \text{Aut}^{\otimes}(w') = H$), the map

$$\text{Aut}^{\otimes}(w')^{\vee}\longrightarrow \text{Aut}^{\otimes}(w)^{\vee}$$ is clearly injective. While I understand how the conclusion follows from this implication, I would love for an explanation of why this is the case, or an alternative argument.

A reference would also be appreciated.

Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor.

Let $(\mathcal{S},w')$ be a full Tannakian sub-category (i.e. closed under tensor products, direct sums, duals and quotients) as well as taking subobjects/subquotients. Denote its fundamental group by $H$.

In "Deligne, Pierre, and J. S. Milne. "Tannakian categories." Lecture Notes in Matehmatics (2012)", Proposition 2.21, it is said that the natural morphism $G\longrightarrow H$ is fully faithful (and in particular surjective). Their proof is tautological, and I could not understand the tautology.

The way the authors deduce the claim is by arguing that the induced map on the underlying algebras (taking the duals of $G = \text{Aut}^{\otimes}(w) \longrightarrow \text{Aut}^{\otimes}(w') = H$), the map

$$\text{Aut}^{\otimes}(w')^{\vee}\longrightarrow \text{Aut}^{\otimes}(w)^{\vee}$$ is clearly injective. While I understand how the conclusion follows from this implication, I would love for an explanation of why this is the case, or an alternative argument.

A reference would also be appreciated.

Let $(\mathcal{T},w)$ be a neutral Tannakian category over a field $k$, with fundamental group $G$, and $w$ a fibre functor.

Let $(\mathcal{S},w')$ be a full Tannakian sub-category (i.e. closed under tensor products, direct sums, duals and quotients) as well as taking subobjects/subquotients. Denote its fundamental group by $H$.

In "Deligne, Pierre, and J. S. Milne. "Tannakian categories." Lecture Notes in Matehmatics (2012)", Proposition 2.21, it is said that the natural morphism $G\longrightarrow H$ is faithfully flat (and in particular surjective). Their proof is tautological, and I could not understand the tautology.

The way the authors deduce the claim is by arguing that the induced map on the underlying algebras (taking the duals of $G = \text{Aut}^{\otimes}(w) \longrightarrow \text{Aut}^{\otimes}(w') = H$), the map

$$\text{Aut}^{\otimes}(w')^{\vee}\longrightarrow \text{Aut}^{\otimes}(w)^{\vee}$$ is clearly injective. While I understand how the conclusion follows from this implication, I would love for an explanation of why this is the case, or an alternative argument.

A reference would also be appreciated.