To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moerdijk "presentation of étendues" where it is stated - unfortunately without proof - at the end of section 2.

I haven't rechecked it in light of the discussion above - but I remembered proving it a few years ago and I was convinced it was correct. Though maybe with all these problems regarding this lemma, it might be time to actually write a proof. In any case, this version of the lemma doesn't have a problem with the example mentioned in the Question.

The comparison lemma in the paper is stated in terms of a functor between two site, so it looks a little different than what is in the elephant, but I think the key modification is that condition (2) is altered in the following way: Both $U$ and $V$ are assumed to be $D$, and the generators of the covering sieve of $U$ (such that the composite with $U \to V$ are in $D$) need to also be themselve in $D$.

The comparison lemma by Kock and Moerdijk has a final assumption of "co-continuity" that might seem a little strange - but if I'm not mistaken it can be deduced (assuming all the other conditions) from the more natural assumption that a Sieve in the domain is a cover if and only if its image by the functor is a cover - that is the functor $u$ is not just "cover preserving" as required by condition (1) but also "cover detecting".

Finally, as the title of the paper suggests, the main motivation of this paper is exactly to prove that "Lemma C.5.2.5" about étendus that Peter is referring to in his answer, which as far as I know is the only use of the comparison lemma to a non-full subcategory that can be found in the Elephant.

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moerdijk "Presentation of étendues" where it is stated — unfortunately without proof — at the end of section 2.

I haven't rechecked it in light of the discussion above — but I remembered proving it a few years ago and I was convinced it was correct. Though maybe with all these problems regarding this lemma, it might be time to actually write a proof. In any case, this version of the lemma doesn't have a problem with the example mentioned in the Question.

The comparison lemma in the paper is stated in terms of a functor between two site, so it looks a little different than what is in the elephant, but I think the key modification is that condition (2) is altered in the following way: Both $U$ and $V$ are assumed to be $D$, and the generators of the covering sieve of $U$ (such that the composite with $U \to V$ are in $D$) need to also be themselve in $D$.

The comparison lemma by Kock and Moerdijk has a final assumption of "co-continuity" that might seem a little strange — but if I'm not mistaken it can be deduced (assuming all the other conditions) from the more natural assumption that a Sieve in the domain is a cover if and only if its image by the functor is a cover — that is the functor $u$ is not just "cover preserving" as required by condition (1) but also "cover detecting".

Finally, as the title of the paper suggests, the main motivation of this paper is exactly to prove that "Lemma C.5.2.5" about étendus that Peter is referring to in his answer, which as far as I know is the only use of the comparison lemma to a non-full subcategory that can be found in the Elephant.

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moerdijk "presentation of étendues" where it is stated - unfortunately without proof - at the end of section 2.

I haven't rechecked it in light of the discussion above - but I remembered proving it a few years ago and I was convinced it was correct. Though maybe with all these problems regarding this lemma, it might be time to actually write a proof. In any case, this version of the lemma doesn't have a problem with the example mentioned in the Question.

The comparison lemma in the paper is stated in terms of a functor between two site, so it looks a little different than what is in the elephant, but I think the key modification is that condition (2) is altered in the following way: Both $U$ and $V$ are assumed to be $D$, and the generators of the covering sieve of $U$ (such that the composite with $U \to V$ are in $D$) need to also be themselve in $D$.

The comparison lemma by Kock and Moerdijk has a final assumption of "co-continuity" that might seem a little strange - but if I'm not mistaken it can be deduced (assuming all the other conditions) by the more natural assumption that a Sieve in the domain is a cover if and only if its image by the functor is a cover - that is the functor $u$ is not just "cover preserving" as required by condition (1) but also "cover detecting".

Finally, as the title of the paper suggests, the main motivation of this paper is exactly to prove that "Lemma C.5.2.5" about étendus that Peter is referring to in his answer, which as far as I know is the only use of the comparison lemma to a non-full subcategory that can be found in the Elephant.

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moerdijk "presentation of étendues" where it is stated - unfortunately without proof - at the end of section 2.

I haven't rechecked it in light of the discussion above - but I remembered proving it a few years ago and I was convinced it was correct. Though maybe with all these problems regarding this lemma, it might be time to actually write a proof. In any case, this version of the lemma doesn't have a problem with the example mentioned in the Question.

The comparison lemma in the paper is stated in terms of a functor between two site, so it looks a little different than what is in the elephant, but I think the key modification is that condition (2) is altered in the following way: Both $U$ and $V$ are assumed to be $D$, and the generators of the covering sieve of $U$ (such that the composite with $U \to V$ are in $D$) need to also be themselve in $D$.

The comparison lemma by Kock and Moerdijk has a final assumption of "co-continuity" that might seem a little strange - but if I'm not mistaken it can be deduced (assuming all the other conditions) from the more natural assumption that a Sieve in the domain is a cover if and only if its image by the functor is a cover - that is the functor $u$ is not just "cover preserving" as required by condition (1) but also "cover detecting".

Finally, as the title of the paper suggests, the main motivation of this paper is exactly to prove that "Lemma C.5.2.5" about étendus that Peter is referring to in his answer, which as far as I know is the only use of the comparison lemma to a non-full subcategory that can be found in the Elephant.