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About every source on string theory I've read which do mention S-duality state that: $$\begin{array}{l} \operatorname S:\operatorname{IIB} \leftrightarrow \operatorname{IIB}\\ \operatorname S:\operatorname{HO} \leftrightarrow \operatorname{I} \end{array}$$
However, how does one prove that the Type IIB string theory is self-S-dual and even more bizzarely, that the Type HO string theory is S-dual to the Type I string theory?
Since S-duality relates a theory at weak coupling to a theory at strong coupling it is in general very hard to rigorously prove that two theories are dual. However, the basic arguments for why it should hold in string theory are given in many text books, see eg chapter 14 in Polchinski or Becker, Becker, Schwarz chapter 8. Here I will just sketch how the relation between type-I and the $SO(32)$ heterotic string theory can be understood.
The first observation is that the massless spectra of the two models agree. Moreover, if we make the identification $$\tag{1} G^I_{\mu\nu} = e^{-\Phi_h} G^h_{\mu\nu} , \qquad \Phi^I = - \Phi^h , \qquad \tilde{F}^I_3 = \tilde{H}^h_3 , \qquad A^I_1 = A^h_1 $$ the low energy effective supergravity actions of the two models match. Since the string coupling constants $g_s^I$ and $g_s^h$ are given as the expectation values of the exponentials of the dilatons $\exp(\Phi^I)$ and $\exp(\Phi^h)$, respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling: $$\tag{2} g^I_s = \frac{1}{g^h_s} . $$ From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by $$\tag{3} l^I_s = l^h_s \sqrt{g^h_s}. $$
As a non-perturbative check we can consider the tension of the type-I D1 brane. The brane is a BPS object, so for all values of the coupling $g_s^I$ the tension is given by the same formula $$ T^I_{D1} = \frac{1}{g_s^I} \frac{1}{2\pi\left(l^I_s\right)^2} = \frac{g^h_s}{2\pi\left(l^h_s\sqrt{g^h_s}\right)^2} = \frac{1}{2\pi\left(l^h_s\right)^2} $$ where I've used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string $$ T^h_{F1} = \frac{1}{2\pi\left(l^h_s\right)^2}. $$ This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string.
There is also a very quick way to see both of these statements, if one is willing to assume that M-theory exists.
In that case, consider M-theory on a fibration with fiber the torus $S^1_A \times S^1_B$. As the radius of $S^1_A$ goes to zero this becomes 10d perturbative type IIA string theory. Then as also the radius of $S^1_B$ goes to zero and we T-dualize it, this becomes 10d perturbative IIB string theory. But the original M-theory did no require any of the two radii to be small, and so if T-duality has an M-theory lift as it should, then M-theory on any such torus should be non-perturbative IIB string theory. That's the picture of F-theory.
Now going through the behaviour of the coupling constants under T-duality (here) one finds that the complexified IIB coupling depends only on the conformal structure of the torus, not on its radii separately. In conclusion then the Moebius group $\mathrm{SL}(2,\mathbb{Z})$ of conformal transformations acting on the torus, which acts by 11d diffeomorphisms and hence ought to be a symmetry of M-theory, acts on IIB states. This is supposed to be the S-duality $IIB \overset{S}{\leftrightarrow} IIB$.
Next, consider the same story but with a $\mathbb{Z}_2 := \mathbb{Z}/2\mathbb{Z}$ orbifolding of the torus included. Take $\mathbb{Z}_2$ to act on $S^1$ by reflection on any line through the center of the circle (when thought of as embedded into $\mathbb{R}^2$ in the canonical way). Write $S^1 // \mathbb{Z}_2$ for the resulting global orbifold.
Now M-theory compactified
on $(S^1_A // \mathbb{Z}_2) \times S^1_B$ yields the heterotic $E_8 \times E_8$ theory
on $S^1_A \times (S^1_B // \mathbb{Z}_2)$ yields type I' string theory.
Both of these statements are originally due to Horava-Witten 95.
But as before, the symmetries of the torus act: Exchanging the two circle factors gives S-duality between heterotic E and type I' string theory. And T-dualizing, as before, turns this into a duality between HO and type I: $$ \array{ HE &\overset{KK/\mathbb{Z}^A_2}{\leftrightarrow}& M &\overset{KK/\mathbb{Z}^B_2}{\leftrightarrow}& I' \\ {T}\updownarrow && && \updownarrow {T} \\ HO && \underset{\phantom{A}S\phantom{A}}{\leftrightarrow} && I } $$
This is what Horava-Witten 95 discuss around their p. 16.
