Counit and Unit to an adjunction

Question

Suppose the diagonal functor $\Delta:\mathcal{C}\to\mathcal{C}^J$ admits both left and right adjoints. How would I describe the units and counits of these adjunctions.

Answer 1

Start with the definition of $\Delta:\mathcal{C}\to\mathcal{C}^J$. This functor takes $c\in \mathcal C$ to the constant functor $\Delta_c:J\to \mathcal C$ (where $\Delta_c(j)=c$ and $\Delta_c(j\overset{f}\to j')=c\overset{1_c}\to c$), and arrows $c\overset{f}\to c'$ to $\Delta_c\overset{\Delta f}\to \Delta_{c'}$, where $\Delta f$ is the natural transformation defined by $\Delta f(i)=f$.

If $\mathcal C$ has small limits, then a good candidate for the right adjoint to $\Delta$ would be $\underset{\leftarrow}\lim:\mathcal C^J\to \mathcal C$ where this functor sends a diagram $D$ in $\mathcal C^J$ to its limit object $d\in \mathcal C.$ Of course, this entails a limit cone $(d,\lambda_i)$ where the $(\lambda_i)$ are arrows from $d$ to $D_i$ that commute with $d_{ij}:D(i)\to D(j).$ On arrows, $\underset{\leftarrow}\lim$ sends natural transformations $\tau:D\to D'$ to the unique arrow $\phi:d\to d'$ that satisfies $\lambda_i\circ \phi=\lambda'_{i}$ for all $i\in J$.

To check that $\Delta\dashv \underset{\leftarrow}\lim$, we look for a natural isomorphism of $\textit{sets},\ \hom(\Delta_ c, D)\leftrightarrow \hom(c,\underset{\leftarrow}\lim D).$

But this is just a matter of interpreting the definitions: an element of the hom set on the left hand side is a natural transformation $\tau$ from $\Delta_c$ to $D$, but by definition of $\Delta,\ \tau$ is nothing more that a cone from $c$ to $D$. And now, by definition of the limit, there corresponds exactly one arrow from $c$ to $\underset{\leftarrow}\lim D$ that satisfies the UMP of the limit. And this is the element of the hom set on the right hand side that we send this cone to. It is routine now to check that this correspondence gives the isomorphism we want.

To find the counit, $\varepsilon_D$ we look for the arrow $\Delta\circ \underset{\leftarrow}\lim D\to D$ that corresponds to the identity arrow in $\text {hom}(\underset{\leftarrow}\lim D,\underset{\leftarrow}\lim D).$ But, the cone from $\underset{\leftarrow}\lim D$ to $D$ that gets sent to the identity arrow in $\hom(\underset{\leftarrow}\lim D,\underset{\leftarrow}\lim D)$ is the limit cone itself so $\varepsilon_D$ is the (natural transfomation defined by) the limit cone.

The unit $\eta,$ of the adjunction is given by $\eta_c:c\to \underset{\leftarrow}\lim \circ \Delta_c=c\times c$ that corresponds to the identity arrow in $\hom(\Delta_c,\Delta_c).$

That the left adjoint to $\Delta$ is the colimit functor is proved in the same way.