Free algebra functor
= Free algebra functor
{title2=$FX=(TX,\mu_X)$}
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 $\mathcal C(X,C)$.