= Left adjoint to the Eilenberg-Moore comparison functor
{c}
If $F\dashv G:\mathcal D\to\mathcal C$ 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 coequalizer
$$
FTA\mathrel{\substack{\xrightarrow{Fa}\\[-2pt]\xrightarrow[\varepsilon_{FA}]{} }}FA\longrightarrow L(A,a).
$$
The pair is reflexive through $F\eta_A$, and its universal property gives $L\dashv K$.
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