Just to be absolutely clear, if $D$ is a linear category, then by $\hat D$ you mean the category of $\mathrm{Ab}$-enriched functors $D^{\mathrm{op}} \to \mathrm{Ab}$. That is, not the full category of $\mathrm{Ab}$-valued presheaves, but only the $\mathrm{Ab}$-enriched ones.

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