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Assume the coequalizers from part (b) exist and write them $q_a:FA\to L(A,a)$. The underlying arrow of the comparison-adjunction unit isWe first relate it to the monad algebra action. Naturality of $\eta$, the coequalizer equation and a monad unit law giveIn $\mathcal{C}$, $a:TA\to A$ is itself a split coequalizer of $Ta,{\mu }_A:T^{2}A\rightrightarrows TA$. Indeed,For an explicit universal-property proof, if $r:TA\to X$ satisfies $rTa=r{\mu }_A$, then $s=r{\eta }_A$ obeysIts uniqueness follows by composing any equation $sa=r$ with ${\eta }_A$.

Now $G{q}_a$ equalizes $Ta$ and ${\mu }_A$. If it is a coequalizer, then $a$ and $G{q}_a$ are coequalizers of the same pair. Their unique comparison map is ${\lambda }_a$, since ${\lambda }_{a}a=G{q}_a$, and is therefore invertible. Conversely, if ${\lambda }_a$ is invertible, $G{q}_a={\lambda }_{a}a$ is an isomorphic copy of the coequalizer $a$, so it is a coequalizer too.

The forgetful functor ${\mathcal{C}}^T\to \mathcal{C}$ reflects isomorphisms: if an monad algebra morphism $u$ has an inverse as an underlying arrow, the equation $ua=bTu$ rearranges to $u^{-1}b=aT(u^{-1})$, making that inverse an monad algebra morphism. Thus underlying invertibility here is exactly invertibility of the unit component. For all monad algebras together, the comparison-adjunction unit criterion isEquivalently, the necessary-and-sufficient conditions for a left adjoint with invertible unit are existence of the coequalizers in part (b) and their preservation by $G$. No preservation of arbitrary coequalizers is required.