I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) (https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf).

My question is similar to this one (Deligne's theorem on the characterisation of Tannakian categories):

Given a cocomplete tensor category $\mathcal{C}$ the paper shows that for every object $X\in \mathcal{C}$ there is a commutative ring $A_X\in \mathcal{C}$ such that $A_X\otimes X\cong A_X^{\oplus\text{dim}X}$, where this is an isomorphism of $A_X$-modules.

However, in the final stage of the proof, it is claimed that there is a commutative ring $B\in \mathcal{C}$ such that $X\otimes B \cong B^{\oplus \text{dim} X}$ holds for all $X\in \mathcal{C}$.

I fail to see how the first statement implies the second. The only idea I have is to consider the coproduct $\bigoplus_{X\in \mathcal{C}} A_X$ over all the $A_X$. But I am not sure if this would even be a commutative ring in $\mathcal{C}$.

I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) in Deligne: Catégories tannakiennes.

My question is similar to this one.

Given a cocomplete tensor category $\mathcal{C}$ the paper shows that for every object $X\in \mathcal{C}$ there is a commutative ring $A_X\in \mathcal{C}$ such that $A_X\otimes X\cong A_X^{\oplus\text{dim}X}$, where this is an isomorphism of $A_X$-modules.

However, in the final stage of the proof, it is claimed that there is a commutative ring $B\in \mathcal{C}$ such that $X\otimes B \cong B^{\oplus \text{dim} X}$ holds for all $X\in \mathcal{C}$.

I fail to see how the first statement implies the second. The only idea I have is to consider the coproduct $\bigoplus_{X\in \mathcal{C}} A_X$ over all the $A_X$. But I am not sure if this would even be a commutative ring in $\mathcal{C}$.

I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) (https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf).

My question is similar to this one (Deligne's theorem on the characterisation of Tannakian categories):

Given a cocomplete tensor category $\mathcal{C}$ the paper shows that for every object $X\in \mathcal{C}$ there is a commutative ring $A_X\in \mathcal{C}$ such that $A_X\otimes X\cong A_X^{\oplus\text{dim}X}$, where this is an isomorphism of $A$-modules.

However, in the final stage of the proof, it is claimed that there is a commutative ring $B\in \mathcal{C}$ such that $X\otimes B \cong B^{\oplus \text{dim} X}$ holds for all $X\in \mathcal{C}$.

I fail to see how the first statement implies the second. The only idea I have is to consider the coproduct $\bigoplus_{X\in \mathcal{C}} A_X$ over all the $A_X$. But I am not sure if this would even be a commutative ring in $\mathcal{C}$.

I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) (https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf).

My question is similar to this one (Deligne's theorem on the characterisation of Tannakian categories):

Given a cocomplete tensor category $\mathcal{C}$ the paper shows that for every object $X\in \mathcal{C}$ there is a commutative ring $A_X\in \mathcal{C}$ such that $A_X\otimes X\cong A_X^{\oplus\text{dim}X}$, where this is an isomorphism of $A_X$-modules.

However, in the final stage of the proof, it is claimed that there is a commutative ring $B\in \mathcal{C}$ such that $X\otimes B \cong B^{\oplus \text{dim} X}$ holds for all $X\in \mathcal{C}$.

I fail to see how the first statement implies the second. The only idea I have is to consider the coproduct $\bigoplus_{X\in \mathcal{C}} A_X$ over all the $A_X$. But I am not sure if this would even be a commutative ring in $\mathcal{C}$.