Left adjoint to the Eilenberg-Moore comparison functor

If $F⊣G:\mathcal{D}\to \mathcal{C}$ induces $T=GF$ and $\mathcal{D}$ has coequalizers of reflexive pairs, then the comparison $K:\mathcal{D}\to {\mathcal{C}}^T$ has a left adjoint. On a $T$-algebra $(A,a)$ it is the coequalizerThe pair is reflexive through $F{\eta }_A$, and its universal property gives $L⊣K$.