Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K \rightarrow \Delta F K$. This is a morphism in $C^J$ and so $F(\eta_K)$ should be well defined. I cannot however compute it using the usual method.

I've been trying to determine what the counit associated with any particular colimit is and decided that its the folding map from a copower of $x$. This is the last thing I need to complete the proof. Any thoughts?