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\$$.