What do you get when you apply a universal cocone to a colimit functor

Question

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?

Answer 1

In general, $\rm colim\circ \Delta$ is the copower by the set $\pi_0(J)$ of connected components of $J$. Thus, $\rm colim(\eta_K)$ is a map ${\rm colim}(K) \to \pi_0(J) \cdot {\rm colim}(K)$, which is determined (like any map out of a colimit) by a cocone consisting of maps $K_j \to \pi_0(J) \cdot {\rm colim}(K)$ for all $j\in J$. At this point there is really only one possible guess for what it might be, namely each $K_j$ maps into the copy of ${\rm colim}(K)$ indexed by the connected component $[j]\in \pi_0(J)$, and into that copy it is the coprojection $K_j \to {\rm colim}(K)$.