Two topos questions

[Peter Arndt <toposopher@gmail.com>]

Hi, category theorists,
 1. In a message to the categories list from 15. jan.1997 (that message can
be seen at http://www.mta.ca/~cat-dist/catlist/1999/finite-topos) Lawvere
talks about "the ... internal topos ... which parametrizes the decidable
K-finites". Does anyone know what exactly is that internal topos? Is there
some morphism that can be seen as the indexed family of decidable K-finites
(just like the generic cardinal "is" the indexed family of finite cardinals
and can be used to construct the full internal subcategory of finite
cardinals)?
 2. An object Y of a topos is said to have locally a property P if there is
an object Z with global support such that Z*(Y) has the property P. For the
topos of sheaves on a T1-space X (and a property P stable under pullback
along subterminals), I convinced myself that this implies the existence of a
covering of X, such that P holds on the restriction of Y to each open set of
the covering. Can this also be proved for schemes or other classes of
topological spaces, maybe with additional conditions on P?
 Thanks a lot!
 Peter