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Apr 2, 2021 at 9:26 comment added Leo Alonso @BertramArnold Why don't you post your comment as an answer? It seems to address the question in full.
Apr 2, 2021 at 9:11 comment added Stabilo Thank you very much @BertramArnold, this is what I was looking for.
Apr 2, 2021 at 9:01 comment added Bertram Arnold The morphism is the composition of applying the functor $-\otimes Y^\vee$ and precomposition with the unit $\mathbf{1}\to Y\otimes Y^\vee$. Thus the middle term in the extension is the pullback $E\otimes Y^\vee\times_{Y \otimes Y^\vee} \mathbf{1}$.
Apr 2, 2021 at 8:22 history asked Stabilo CC BY-SA 4.0