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.
Perhaps you could include the diagram from your notes in your question. It might be easier to guess from there. What comes to my mind is to apply $GF$ to the triangular equation $G\varepsilon \circ \eta G = 1_G$ and evaluate the resulting natural maps at the object $B$. This would give $GFG\varepsilon_B \circ GF\eta_{GB} = 1_{GFGB}$. –  Marc Olschok Apr 25 '12 at 17:26