  The Schur-Weyl duality, which started as the study of the commuting actions
of the symmetric group $S_d$ and $\mathrm{GL}(n,\mathbb{C})$ on $V^{\otimes d}$
where $V=\mathbb{C}^n$, was extended by Drinfeld and Jimbo to the context of
the finite Iwahori-Hecke algebra $H_d(q^2)$ and quantum algebras
$U_q(\mathrm{gl}(n))$, on using universal $R$-matrices, which solve the
Yang-Baxter equation. There were two extensions of this duality in the
Hecke-quantum case: to the affine case, by Chari and Pressley, and to the super
case, by Moon and by Mitsuhashi. We complete this chain of works by completing
the cube, dealing with the general affine super case, relating the commuting
actions of the affine Iwahori-Hecke algebra $H^a_d(q^2)$ and of the affine
quantum Lie superalgebra $U_{q,a}^\sigma(\mathrm{sl}(m,n))$ using the
presentation by Yamane in terms of generators and relations, acting on the
$d$th tensor power of the superspace $V=\mathbb{C}^{m+n}$. Thus we construct a
functor and show it is an equivalence of categories of $H_d^a(q^2)$ and
$U_{q,a}^\sigma(\mathrm{sl}(m,n))$-modules when $d<n'=m+n$.
