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I think there’s no less generality from taking $D=\mathbf{Set}$. If no-one objects, I suggest to replace $D$ with $\mathbf{Set}$ and then put a remark about this at the end of the article. The point is that the definition remains the same if you compose $F:C\to D$ with the global sections functor $D(I,-):D \to \mathbf{Set}$, which is monoidal.
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