Free functor into a Kleisli category
= Free functor into a Kleisli category
{title2=$F_T$}
The free functor $F_T:\mathcal C\to\mathcal C_T$ is the identity on objects and sends $f:A\to B$ to $\eta_Bf:A\to TB$. A functor $G:\mathcal C\to\mathcal D$ carries an algebra for the precomposition monad $G\mapsto GT$ exactly when it factors through $F_T$.