%
\label{prop:ch42-holend-exponentials}
For any $\tau$-objects $A, B, C$,
the internal Hom $[A,B]$
\textup{(}Definition~\textup{\ref{def:hom-object},} II.D54\textup{)}
satisfies the exponential law:
\[
    \boxed{%
    [A \times B,\; C]
    \;\cong\;
    [A,\; [B, C]]}
\]
as $\tau$-objects,
where $\times$ is the Cartesian product in~$\tau$.

[Proof sketch]
The isomorphism is built stagewise.
At each finite stage~$k$
of the primorial tower,
the category $\tau_k$
is a finite category
with finite Hom sets.
The standard exponential law
for finite sets gives
\[
    \Hom_{\tau_k}(A_k \times B_k,\; C_k)
    \;\cong\;
    \Hom_{\tau_k}\bigl(A_k,\; \Hom_{\tau_k}(B_k, C_k)\bigr)
\]
naturally in $A_k$, $B_k$, $C_k$.
This bijection sends
$f_k : A_k \times B_k \to C_k$
to the curried map
$\hat{f}_k : A_k \to \Hom_{\tau_k}(B_k, C_k)$
defined by $\hat{f}_k(a)(b) = f_k(a, b)$.

The connecting maps $r_{k+1,k}$
of the primorial tower
are compatible with currying:
if $f_{k+1}$ reduces to $f_k$ modulo $P_k$,
then $\hat{f}_{k+1}$ reduces to $\hat{f}_k$ modulo $P_k$.
This compatibility follows from the functoriality
of the reduction maps
(they are ring homomorphisms,
hence preserve the product structure).

Passing to the inverse limit:
\[
    [A \times B,\; C]
    \;=\;
    \varprojlim_k \Hom_{\tau_k}(A_k \times B_k, C_k)
    \;\cong\;
    \varprojlim_k \Hom_{\tau_k}(A_k, \Hom_{\tau_k}(B_k, C_k))
    \;=\;
    [A,\; [B, C]].
\]
The isomorphism respects
the bipolar decomposition (II.P11)
because currying commutes
with the idempotent projections $e_\pm$:
$\widehat{e_\pm f} = e_\pm \hat{f}$.