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The converse seems to hold, the functor $_pu$ preserving sheaves implies $u$ is cocontinuous [SGA 4, Exp. III, Proposition 2.2].

As every DVR is a Noetherian local ring of dimension 1, all DVRs are J-2 rings (by (4)). Therefore, would it not make more sense to remove the condition on the characteristic in (6)? Of course the characteristic 0 hypothesis is relevant when discussing excellent rings, but this lemma is about a weaker property.

As every DVR is a Noetherian local ring of dimension 1, all DVRs are J-2 rings (by (4)). Therefore, would it not make more sense to remove the condition on the characteristic in (6)? Of course the characteristic 0 hypothesis is relevant when discussing excellent rings, but this lemma is about a weaker property.

This result can be strengthened. Namely, if $u$ preserves fibre products, then the following are equivalent [SGA4, Exp. III, Proposition 1.6]:

1. For every $\mathcal{V}=\{V_i\to V\}$ in $\operatorname{Cov}(\mathcal{C})$ we have that the sieve generated by $u(\mathcal{V})=\{u(V_i)\to u(V)\}$ (Definition 7.47.3) is a covering sieve in $\mathcal{D}$. In other words, there is a cover $\mathcal{U}$ in $\mathcal{D}$ refining $u(\mathcal{V})$.

2. $u^p\mathcal{F}$ is a sheaf on $\mathcal{C}$ for every sheaf $\mathcal{F}$ on $\mathcal{D}$.

(This is the dual comment to #11579.)

For the reader's benefit, I want to record that, given a functor $u:\mathcal C\to\mathcal D$ and a presheaf $\mathcal F:\mathcal C^{\mathrm{opp}}\to\mathrm{Sets}$, the presheaf ${}_p u\mathcal F:\mathcal D^{\mathrm{opp}}\to\mathrm{Sets}$ is the right Kan extension of $\mathcal F$ along $u^{\mathrm{opp}}$. The formula $${}_p u\mathcal (F)(V)=\lim_{({}_V\mathcal I)^{\mathrm{opp}}}{}_V\mathcal F$$ is exactly [Rie, eq. (6.2.3)], and Lemma 7.19.2 is the right-adjoint assertion in [Rie, Proposition 6.1.5].

References

[Rie] Emily Riehl, Category Theory in Context

Suggested slogan: the left Kan extension of a representable presheaf is representable.

In the second proof of smooth base change, in the last paragraph you invoke Lemma 0A5E which assumes properness, I think we should rather use point (6) of Theorem 03SC (the way I see it, this is the crucial point where "torsion is invertible on the base" is really needed).

Alternative terminology proposal: "A functor that satisfies (the) comparison (lemma)" or "a comparison functor": Lemma 7.29.1 is just a form of the comparison lemma.

In the proof: The last sentence of the first paragraph seems to be incomplete.

Suggested fix: The second paragraph explains the construction of $X$. Remove the last sentence of the first paragraph and add "It is then clear that $P_{\bullet} = X(A \to B)$." to the second paragraph

It is clear from context, but maybe you should mention that $\kappa=R/\mathfrak{m}$ for clarity.

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typo: in the the very last displayed composite, the middle $b$ should be $b'$ and $bot$ and $top$ are swapped. That it, is rather should be: $$k\circ a'\circ b'\xrightarrow{\xi_{top}}a\circ l_*\circ b'\xrightarrow{\xi_{bot}}a\circ b\circ m_*.$$

at the end of the proof and also above in the calculation $d^{n-1} \circ h^{n-1}$ is considerd. I assume there is a wrong index here since $h^{n-1}$ goes from $K^{n-1} \to I^{n-2}$ meaning that you cannot apply $d^{n-1}$ now.

About remark 0GU2. Shouldn't the inclusion of locally finite direct sum and direct sum be reversed? Or the direct sum replaced with direct product?

@#10213: Yes, this is confirmed e.g. in the proof of Set Theory, Lemma 3.11.1: “Note that two families $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$ and $\mathcal{V} = \{V_j \to V\}_{j \in J}$ are combinatorially equivalent if and only if $Supp(\mathcal{U}) = Supp(\mathcal{V})$.”

The point is that the definition of a family of morphisms with fixed target, Definition 7.6.1, includes the indexing set $I$ as part of the data. This gives the flexibility to allow repetition of the same morphism, for instance $\mathcal{U}=\{U\xrightarrow{\operatorname{id}} U\}$ and $\mathcal{V}=\coprod_{n\in\mathbb{N}}\{U\xrightarrow{\operatorname{id}}U\}$ are two distinct families of morphisms with fixed target that are combinatorially equivalent. Definition 7.6.1 is the notion that is required to define a pretopology in the sense of SGA 4, Exp. II, Definition 1.3, named simply as “site” here in the Stacks Project, Definition 7.6.2.

Definition 7.6.2 is to Definition 7.6.1 what a topology (Definition 7.47.6) is to a sieve (Definition 7.47.1). The notion of a sieve does not allow for repetition of morphisms.

Regarding the very first sentence: "Suppose that $\mathcal{C}$ is a category (as in Categories, Definition 4.2.1) and that $\text{Cov}(\mathcal{C})$ is a proper class of coverings satisfying properties (1), (2), and (3) of Sites, Definition 7.6.2." The referred definitions specify $\operatorname{Mor}(\mathcal{C})$ is a set and that also $\operatorname{Cov}(\mathcal{C})$ is a set. Shouldn't one replace then "proper class of coverings" by "set of coverings"?

In the comment after Lemma 6.21.2 : It doesn't make sense to say that $f_p G$ is a left adjoint, it's a sheave, not a functor. It is $f_p$ that is adjoint.

Thanks for adding the correction!

Alternative to #11494, another reference for the sharpened form of the result (yielding an actual adjointness in the $2$-category of triangulated categories) is [lipman-notes, Proposition 3.3.8].

Typo: at the beginning, after "It follows from Derived Categories, Lemma 13.31.7 that", in condition (1), replace "trianguated" by "triangulated."