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Let's consider the bicategories, LogTopos of elementary topoi, logical functors and natural transformations and GrTopos of Grothendieck topoi, geometric morphisms and natural transformations.

The standard free topoi arise from left-adjoints to functors from the 1-category part of LogTopos to category of graphs, if I understand correctly.

We have the obvious forgetful pseudofunctors from LogTopos/GrTopos, for GrTopos 1 for each direct/inverse image, into Cat. What is known about existence of left/right adjoints to these pseudofunctors?

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    $\begingroup$ By bifunctor, do you mean a ncatlab.org/nlab/show/… ? $\endgroup$
    – David Roberts ♦
    Commented Aug 25 at 8:58
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    $\begingroup$ This is a difficult question and you haven't got any answers yet, so I'll throw in some guesses to start the discussion: you may want the base 2-category to be Lex (categories with finite limits); then the free Grothdieck topos should be something like the Yoneda embedding and the free logical topos something like the von Neumann hierarchy; there probably aren't any right adjoints. $\endgroup$
    – Paul Taylor
    Commented Aug 26 at 9:18
  • $\begingroup$ @PaulTaylor What do you mean by the "like the von Neumann hierarchy" part? My current guess is approaching it through, Dostál's Pseudoadjoint Functor Theorem. But that form gives a recognition of a pseudoadjoint pair of pseudofunctors rather than a recognition of left/right pseudoadjoint, so it will likely need some massaging to get into the correct form. $\endgroup$
    – Ilk
    Commented Aug 26 at 10:38
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    $\begingroup$ It's supposedly well-known that the 2-category of toposes and logical morphisms is 2-monadic over categories, which in particular involves a left adjoint answering part of your question. For instance you can see this claimed here: ncatlab.org/nlab/show/topos I'm realizing I don't know a reference, though; Blackwell, Kelly, and Power don't get all the way there in their canonical paper on 2-monads. $\endgroup$
    – Kevin Carlson
    Commented Aug 27 at 3:31
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    $\begingroup$ Right, sorry about my misreading; it's actually well-known that toposes are monadic over categories and natural isomorphisms between functors. It's apparently also known that, say, cartesian closed categories are not monadic over the full 2-category of categories you're looking for and I would guess that however that works also rules this out for toposes. $\endgroup$
    – Kevin Carlson
    Commented Aug 27 at 23:33

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In one of the directions suggested by Paul Taylor in the comments, by Diaconescu's theorem the forgetful functor from Grothendieck toposes and inverse image functors to CAT (the category of large categories) has a partial left adjoint defined on all small categories, which sends a small category $C$ to the presheaf category $[(FC)^{\rm op},\rm Set]$, where $FC$ is the free category-with-finite-limits generated by $C$ (which is also small). The latter can be constructed concretely, up to equivalence, as the closure of $C$ under finite limits in its co-presheaf category $[C,\rm Set]^{\rm op}$, so the whole thing is sort of a double Yoneda embedding.

And as Kevin Carlson said in the comments, the forgetful functor from elementary toposes, logical morphisms, and natural isomorphisms to the category of categories, functors, and natural isomorphisms has a left adjoint.

I doubt that any of the other adjoints you're wondering about exist.

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