  This paper is the third in a series whose goal is to develop a fundamentally
new way of viewing theories of physics. Our basic contention is that
constructing a theory of physics is equivalent to finding a representation in a
topos of a certain formal language that is attached to the system. In paper II,
we studied the topos representations of the propositional language PL(S) for
the case of quantum theory, and in the present paper we do the same thing for
the, more extensive, local language L(S). One of the main achievements is to
find a topos representation for self-adjoint operators. This involves showing
that, for any physical quantity A, there is an arrow
$\breve{\delta}^o(A):\Sig\map\SR$, where $\SR$ is the quantity-value object for
this theory. The construction of $\breve{\delta}^o(A)$ is an extension of the
daseinisation of projection operators that was discussed in paper II. The
object $\SR$ is a monoid-object only in the topos, $\tau_\phi$, of the theory,
and to enhance the applicability of the formalism, we apply to $\SR$ a topos
analogue of the Grothendieck extension of a monoid to a group. The resulting
object, $\kSR$, is an abelian group-object in $\tau_\phi$. We also discuss
another candidate, $\PR{\mathR}$, for the quantity-value object. In this
presheaf, both inner and outer daseinisation are used in a symmetric way.
Finally, there is a brief discussion of the role of unitary operators in the
quantum topos scheme.
