  This is the third paper of this series. In \cite{Wang20}, we defined the
monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact
oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed
2-form viewed as a decoration. In this paper, we establish a gluing theorem for
this Floer homology when two such 3-manifolds are glued suitably along their
common boundary, assuming that $\partial Y$ is disconnected, and $\omega$ is
small and yet non-vanishing on $\partial Y$.
  As applications, we construct a monopole Floer 2-functor and the generalized
cobordism maps. Using results of Kronheimer-Mrowka and Ni, it is shown that for
any such 3-manifold $Y$ that is irreducible, this Floer homology detects the
Thurston norm on $H_2(Y,\partial Y;\mathbb{R})$ and the fiberness of $Y$.
Finally, we show that our construction recovers the monopole link Floer
homology for any link inside a closed 3-manifold.
