Christopher Upshaw wrote:

>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 : m\to g\$$ \$f(T) = f (TT) =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\$$.

I don't understand your reasoning here. In which category does \$$f\$$ live in? And why are we using \$$T\$$?