Free algebra functor ($FX=(TX,{\mu }_X)$)

For a monad $(T,\eta ,\mu )$, the free algebra functor $F:\mathcal{C}\to {\mathcal{C}}^T$ sends $X$ to $(TX,{\mu }_X)$ and $f$ to $Tf$. It is a left adjoint to the forgetful functor $U$, via $h\mapsto h{\eta }_X$ and $u\mapsto \gamma Tu$ between ${\mathcal{C}}^T(FX,(C,\gamma ))$ and $\mathcal{C}(X,C)$.