What is the algebras of the free monoid monad?

Question

Consider the forgetful $\vdash$ free adjunction induced by the free functor $F:Set\to Mon$ and forgetful functor $U:Mon\to Set$. Define $T=UF$. Also, define $\eta:id_{Set}\Rightarrow T $ by $\eta_X(x)=[x]$. Finally, define $\mu:T^2\Rightarrow T$ by $$\mu_X([x_{11},\cdots,x_{1n_{1}}],\cdots,[x_{m1},\cdots,x_{mn_{m}}])=[x_{11},\cdots,x_{mn_{m}}].$$ Note that the algebras must be pairs $(X,\xi_X)$ where $\xi_X:TX\to X$ is defined by $\xi_{X}([a])=a$ and $$\xi_X(\xi_X([x_{11},\cdots,x_{1n_{1}}]),\cdots\xi_X([x_{m1},\cdots,x_{mn_{m}}]))=\xi_X([x_{11},\cdots,x_{mn_{m}}]).$$ What else can I say about the algebras? Also, how would I show that the algebra morphisms are precisely the homomorphisms?