  We compare the (horizontal) trace of the affine Hecke category with the
elliptic Hall algebra, thus obtaining an "affine" version of the construction
of [14]. Explicitly, we show that the aforementioned trace is generated by the
objects $E_{\textbf{d}} = \text{Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 \dots
T_{n-1})$ as $\textbf{d} = (d_1,\dots,d_n) \in \mathbb{Z}^n$, where $Y_i$
denote the Wakimoto objects of [9] and $T_i$ denote Rouquier complexes. We
compute certain categorical commutators between the $E_{\textbf{d}}$'s and show
that they match the categorical commutators between the sheaves
$\mathcal{E}_{\textbf{d}}$ on the flag commuting stack, that were considered in
[27]. At the level of $K$-theory, these commutators yield a certain integral
form $\widetilde{\mathcal{A}}$ of the elliptic Hall algebra, which we can thus
map to the $K$-theory of the trace of the affine Hecke category.
