= Comparison-adjunction unit criterion
{title2=$\lambda_{(A,a)}\text{ invertible}\iff Gq_a\text{ is a coequalizer}$}
For $F\dashv G$ inducing $T=GF$, the <Eilenberg-Moore comparison functor> has a <left adjoint> exactly when every algebra $(A,a)$ has a <coequalizer> $q_a:FA\to L(A,a)$ of $Fa,\varepsilon_{FA}:FTA\rightrightarrows FA$. Transposition identifies maps equalizing this pair with <monad algebra morphisms> $(A,a)\to K(D)$, proving both directions of the criterion. The unit is $\lambda=Gq_a\eta_A$ and satisfies $Gq_a=\lambda a$. In the base category, $a:TA\to A$ is a <split coequalizer> of $Ta,\mu_A:T^2A\rightrightarrows TA$, split by $\eta_A$ and $\eta_{TA}$. Consequently $\lambda$ is invertible exactly when $Gq_a$ is also a coequalizer. The unit is invertible everywhere precisely when $G$ preserves all these specified coequalizers; preservation of arbitrary coequalizers is not required.
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