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In

http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf

on p55 the following three facts are stated without proof:

Every topos is a regular category;

Every topos has finite colimits, and the initial object is strict;

In every topos, every operation $$\phi^{\sharp}:\mathrm{Sub}(X)\rightarrow \mathrm{Sub}(Y)$$ along $\phi:X\rightarrow Y$ has a right adjoint.

Could someone possibly help me with the proof of these three statements, particularly the first?

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A good general reference for such facts is Peter Johnstone's book "Topos Theory" but I think these results were also proved in Peter Freyd's 1972 article "Aspects of Topoi".

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    $\begingroup$ A more recent reference (1992) that can also help is "Sheaves in geometry and logic" by MacLane and Moerdijk. They prove these facts both in case of Grothendieck topoi and also in the general setting using the axioms of topoi. $\endgroup$
    – godelian
    Commented Feb 18, 2011 at 21:40

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