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\\).

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\\).