The internal logic of a topos

[Vaughan Pratt <pratt@cs.stanford.edu>]

As I understand the internal logic of a topos it consists of certain
morphisms from finite powers of Omega to Omega.  In the case of Set it
consists of all such morphisms.  For which toposes is this not the case,
and for those how are the morphisms that do belong to the internal logic
best characterized?

I do hope it's not necessary to start from the notion of an internal
Heyting algebra, that sounds so counter to mathematical practice and
intuition.

If the internal logic consists of precisely those morphisms preserved by
geometric morphisms this will give me the necessary motivation to go to
the mats with geometry.

Vaughan