categories: injective modules in a topos

[Michael Barr <barr@barrs.org>]

A week or so ago, I asked the question about injectives in a category of
modules in over a ring object in a Grothendieck topos.  I asked whether if
I is an injective module and E is an object of the topos, I^E is
injective.  I got no useful answers.  Here is a related question.  Does
anyone know if an injective is interally injective?  That is, if A and B
are modules, then there is an object of the topos A -o B that is the
subobject of B^A consisting of the additive morphisms.  So what I am
asking is whether for an injective I, the induced B -o I --> A -o I is
epic.  Or rather, can every module be embedded into an internal injective?
Is there an internally injective cogenerator?

Michael