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The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps it only works in my setting, pullbacks along

  • (surjective ?) relatively tidy geometric morphism between (localic) toposes or equivalently for locales,
  • a surjective semi-proper/lax proper/perfect locale arrow between locales

knowing that, for locales: a lax kernel pair arises as usual as the lax kernel pair of its lax coequalizer [necessarily semi-proper and surjective], when at least one of the two projections is proper, which is always the case in my personal setting. [The generalization for toposes is likely to be the same where we replace semi-proper by relatively tidy]

I recall a lax kernel pair, in general category, of an arrow $f$ is the universal data of some arrows $p_1$ and $p_2$ such that there is a 2-cell $\eta \colon f \circ p_1 \Rightarrow f \circ p_2$.

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