Lemma 21.30.2. In Situation 21.30.1 for $X$ in $\mathcal{C}$ denote $\mathcal{A}_ X$ the objects of $\textit{Ab}(\mathcal{C}_\tau /X)$ of the form $\epsilon _ X^{-1}\mathcal{F}'$ with $\mathcal{F}'$ in $\mathcal{A}'_ X$. Then

1. for $\mathcal{F}$ in $\textit{Ab}(\mathcal{C}_\tau /X)$ we have $\mathcal{F} \in \mathcal{A}_ X \Leftrightarrow \epsilon _{X, *}\mathcal{F} \in \mathcal{A}'_ X$, and

2. $f_\tau ^{-1}$ sends $\mathcal{A}_ Y$ into $\mathcal{A}_ X$ for any morphism $f : X \to Y$ of $\mathcal{C}$.

Proof. Part (1) follows from (3) and part (2) follows from (2) and the commutativity of (21.30.0.1) which gives $\epsilon _ X^{-1} \circ f_{\tau '}^{-1} = f_\tau ^{-1} \circ \epsilon _ Y^{-1}$. $\square$

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