Every topos is a regular category

Question

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?

Answer 1

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".