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I'm trying to understand sections [61.25] and [61.26] of Stacks Project on closed immersions and extension by zero on the pro-étale site.

Lemma [61.25.5] refers to affine weakly contractible objects $U \rightarrow X$ such that, with $i:Z\rightarrow X$ a closed immersion, every point of $U$ specializes to a point of $U \times_XZ$. I understand that, because closed immersions are stable under base change, $U \times_X Z$ identifies with an affine closed subscheme of $U$. Further, [61.26.5] takes an affine w-contractible $V \rightarrow X$ and it is implied that it verifies the preceding condition.

I really fail to see why this property of specialising to the intersection holds for w-contractile (and hence w-local) objects and I feel like I'm missing something obvious. Any (partial) explanation or clarification is most welcome.

Thank you in advance

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This post was born out of a very idiotic error.

A friend highlighted a counterexample : take $X$ the disjoint union of two copies of V w-contractible affine, identify V with a copy and take Z to be the other copy. Clearly this fails.

In fact I had wrongly read 61.26.5 where $\tilde{V}$ is precisely where things specialize as wanted...

Feel free to delete

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