A (possibly) large site is a small-generated site if it has a small dense sub-site.
In (Johnstone, p. 548) this is called an essentially small site. But notice that the underlying category of such need not be an essentially small category.
The category of paracompact manifolds with its standard open cover coverage is a large site that is small-generated: a dense sub-site is CartSp${}_{top}$.
Similarly the category Diff of smooth manifolds with the open cover coverage is a large site that is small-generated: a dense sub-site is CartSp${}_{smooth}$.
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