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Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences?

To give this some context, I'd like to extract a (cofibrant, if possible) simplicial set from the Waldhausen S-construction applied to a category with cofibrations, and I realized that my standard way of taking a nerve is for simplicial categories (i.e. simplicial objects in categories for which the objects form a constant simplicial set), and this doesn't clearly apply to the S-construction.

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# Nerves of simplicial objects in categories/Waldhausen's S-construction

Is there a good nerve-like functor from simplicial objects in categories to simplicial sets which takes level-wise equivalences of categories to weak equivalences?

To give this some context, I'd like to extract a (cofibrant, if possible) simplicial set from the Waldhausen S-construction applied to a category with cofibrations, and I realized that my standard way of taking a nerve is for simplicial categories (i.e. simplicial objects in categories for which the objects form a constant simplicial set), and this doesn't clearly apply to the S-construction.