I don't think \\(\mathbf{Grp}[L(m),g] \cong \mathbf{Mon}[m,R(g)]\\) is true with \\(\mathbf{L} = \mathbf{T}\circ\mathbf{S}\\).

Let's look at \\(m = (bool, or), g = L(m)\\).

Given \\(f \in [m, g]\\) \\[f(t) = f (t+t) =f(t)f(t)\\] but the only element of g with that property is \\(\epsilon\\). So there is exactly one morephism on the rhs.

But on the lhs, there are at least two, \\(x\mapsto \epsilon\\) and \\(x \mapsto x\\).