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Let $T$ be an abelian rigid monoidal category and $\mathbf{1}$ be a unit object in $T$. For two objects $X$ and $Y$ in $T$, there is a natural group isomorphism
$$n:\operatorname{Ext}^1_{T}(Y,X)\stackrel{\sim}{\longrightarrow}\operatorname{Ext}^1_{T}(\mathbf{1},X\otimes Y^{\vee})$$ induced by $\operatorname{Hom}_T(Y,X)\cong \operatorname{Hom}_T(\mathbf{1},X\otimes Y^{\vee})$. Can one describe $n$ explicitely in terms of Yoneda extensions? Namely, what is "$?$" in $$n(0\to X \to E \to Y \to 0)~\equiv ~0\to X\otimes Y^{\vee}\to ~?~ \to \mathbf{1} \to 0 $$ where $\equiv$ determines extensions up to congruence relations.

Many thanks!

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    $\begingroup$ 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 $E\otimes Y^\vee\times_{Y \otimes Y^\vee} \mathbf{1}$. $\endgroup$
    – Bertram Arnold
    Commented Apr 2, 2021 at 9:01
  • $\begingroup$ Thank you very much @BertramArnold, this is what I was looking for. $\endgroup$
    – Stabilo
    Commented Apr 2, 2021 at 9:11
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    $\begingroup$ @BertramArnold Why don't you post your comment as an answer? It seems to address the question in full. $\endgroup$
    – Leo Alonso
    Commented Apr 2, 2021 at 9:26

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