Let $F: \mathcal{C} \rightarrow \mathcal{D}$ be left adjoint to $G: \mathcal{D} \rightarrow \mathcal{C}$. Consider the unit $\eta$ and counit $\varepsilon$ of this adjunction.
Is it true that $GFG \varepsilon_B \circ \eta_{GFGB} = 1_{GFGB}$?
There is a split co-equaliser diagram in my notes, but I suspect that a couple of the arrows might be the wrong way round - the above morphism is equal to $\eta_{GB} \circ G\varepsilon_B$, which is backwards from the $\Delta$ identities. Any help would be appreciated.