If is a full subcategory then the second condition is automatic.
The following theorem is known as the comparison lemma.
This appears as (Johnstone, theorem C2.2.3).
2’. For every morphism in with there is a cover in generated by a family of morphisms in such that the composites are in .
But this is too weak to prove the comparison lemma as the following example shows:
Let be any groupoid, with the trivial topology (only maximal sieves cover), and let be the discrete category on the same objects. Then for any morphism , its inverse generates the maximal sieve on , and the composite is in , so the conditions 1 and 2’ of the definition are satisfied. But the restriction is not generally an equivalence.
See the dicussion here.
Let be a locale with frame regarded as a site with the canonical coverage ( covers if the join of the is ). Let be a basis for the topology of : a complete join-semilattice such that every object of is the join of objects of . Then is a dense sub-site.
For the category of all topological manifolds equipped with the open cover coverage, the category CartSp is a dense sub-site: every topological manifold has an open cover by open balls homeomorphic to a Cartesian space.
For the category of all paracompact topological manifolds equipped with the good open cover coverage, the category CartSp is a dense sub-site: every paracompact topological manifold has an good open cover by open balls homeomorphic to a Cartesian space.
Similarly for Diff the category of paracompact smooth manifolds equipped with the good open cover coverage, the full subcategory CartSp is a dense sub-site: every such smooth manifold has a differentiably good open cover (see there): a good cover by open balls each of which are diffeomorphic to a Cartesian space.
The comparison lemma originates with the exposé III by Verdier in
A more general form is used to give a site characterization for étendue toposes in
A proof of the comparison lemma together with a nice list of examples is in