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Zhen Lin's answer to the MSE question mentions that $\mathsf{Sh}(-)$ is a stack for the canonical topology on the site of open subsets of a space. One of the comments asks whether the stack descent condition can be realized as a strict 1-limit. Finally, Zhen Lin writes this is doable, but for a different diagram (though of the same shape).

What are the details? Is this part of a more general story?

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    $\begingroup$ If you want some keywords, look up Reedy fibrant diagrams and homotopy limits. $\endgroup$
    – Zhen Lin
    May 21, 2016 at 17:18

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