When explaining how Heyting categories can model first order logic it would be nice to be able to give some small example and contrast it with Set-semantics. I realized however that I don't know of any Heyting category which is not also a topos. It would be nice to have more concrete examples to give.

A search netted me a [master thesis][1] that discussed the Heyting category structure on some categories of graphs. Do you have other examples of Heyting categories that are not toposes?

EDIT: Syntactic categories are Heyting categories which are not necessarily toposes as pointed out by godelian. I am however looking for examples not arising out of logic (:


  [1]: https://scholarworks.umt.edu/etd/1124/