Let $\mathcal{C}$ be a locally small category and let $\mathcal{J}$ be a small category. Assume that $\mathcal{C}$ has all $\mathcal{J}$-shaped limits and colimits. Describe the unit and counit for the adjunctions between the constant diagram functor $\Delta: \mathcal{C} \rightarrow \mathcal{C}^{\mathcal{J}}$ and the limit and colimit functors $\mathcal{C}^{\mathcal{J}} \rightrightarrows \mathcal{C}$.
The only thing that has come to mind is to use the fact that $\mathcal J$ is small and therefore any limit can be written out in terms of products and equalizer. But even then it remains unclear how this helps determine the unit. Same goes for the counit.
Thanks in advanced for any help.
I will do the case for the limit, the colimit is analogous. We have that $lim: C^J \rightarrow C$ is right adjoint to the constant functor $\Delta: C \rightarrow C^J$. Explicitly, for $x \in C$, we have a natural bijection $$C^J(\Delta(x),(-)) \cong C(x,lim(-)) $$ given by the universal property of the limit. The unit of this adjunction is given by the element $\alpha_x \in C(x,lim(\Delta(x)))$ which represents the natural transformation $C^J(\Delta(x),(-)) \cong C(x,lim(-))$ since $C^J(\Delta(X),(-))$ is representable. Now, note that, the $lim(\Delta(x))=x$ and the bijection simply takes he identity of functor $id_{\Delta(x)}: \Delta(x) \rightarrow \Delta(x)$ (which is the identity in $x$) to the identity in $x$, $id_x$. Thus we conclude that the unit of this adjunction is simply the identity ($id_{id_C}: id_C \rightarrow id_C$).