  Motivated by spectral gluing patterns in the Betti Langlands program, we show
that for any reductive group $G$, a parabolic subgroup $P$, and a topological
surface $M$, the (enhanced) spectral Eisenstein series category of $M$ is the
factorization homology over $M$ of the $\mathrm{E}_2$-Hecke category
$\mathrm{H}_{G, P} = \mathrm{IndCoh}(\mathrm{LS}_{G, P}(D^2, S^1))$, where
$\mathrm{LS}_{G, P}(D^2, S^1)$ denotes the moduli stack of $G$-local systems on
a disk together with a $P$-reduction on the boundary circle.
  More generally, for any pair of stacks $\mathcal{Y}\to \mathcal{Z}$
satisfying some mild conditions and any map between topological spaces $N\to
M$, we define $(\mathcal{Y}, \mathcal{Z})^{N, M} = \mathcal{Y}^N
\times_{\mathcal{Z}^N} \mathcal{Z}^M$ to be the space of maps from $M$ to
$\mathcal{Z}$ along with a lift to $\mathcal{Y}$ of its restriction to $N$.
Using the pair of pants construction, we define an $\mathrm{E}_n$-category
$\mathrm{H}_n(\mathcal{Y}, \mathcal{Z}) =
\mathrm{IndCoh}_0\left(\left((\mathcal{Y}, \mathcal{Z})^{S^{n-1},
D^n}\right)^\wedge_{\mathcal{Y}}\right)$ and compute its factorization homology
on any $d$-dimensional manifold $M$ with $d\leq n$, \[
  \int_M \mathrm{H}_n(\mathcal{Y}, \mathcal{Z}) \simeq
\mathrm{IndCoh}_0\left(\left((\mathcal{Y}, \mathcal{Z})^{\partial (M\times
D^{n-d}), M}\right)^\wedge_{\mathcal{Y}^M}\right), \] where $\mathrm{IndCoh}_0$
is the sheaf theory introduced by Arinkin--Gaitsgory and Beraldo. Our result
naturally extends previous known computations of Ben-Zvi--Francis--Nadler and
Beraldo.
