We have a quadruple adjunction
\begin{align*} \textup{Hom}_{\mathsf{Sets}}\webleft (\pi _{0}\webleft (\mathcal{C}\webright ),X\webright ) & \cong \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},X_{\mathsf{disc}}\webright ),\\ \textup{Hom}_{\mathsf{Cats}}\webleft (X_{\mathsf{disc}},\mathcal{C}\webright ) & \cong \textup{Hom}_{\mathsf{Sets}}\webleft (X,\text{Obj}\webleft (\mathcal{C}\webright )\webright ),\\ \textup{Hom}_{\mathsf{Sets}}\webleft (\text{Obj}\webleft (\mathcal{C}\webright ),X\webright ) & \cong \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},X_{\mathsf{indisc}}\webright ), \end{align*}
natural in $\mathcal{C}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, where
- The functor
\[ \pi _{0}\colon \mathsf{Cats}\to \mathsf{Sets}, \]
the connected components functor, is the functor sending a category to its set of connected components of Definition 9.3.2.2.1.
- The functor
\[ \webleft (-\webright )_{\mathsf{disc}}\colon \mathsf{Sets}\to \mathsf{Cats}, \]
the discrete category functor, is the functor sending a set to its associated discrete category of Item 1.
- The functor
\[ \text{Obj}\colon \mathsf{Cats}\to \mathsf{Sets}, \]
the object functor, is the functor sending a category to its set of objects.
- The functor
\[ \webleft (-\webright )_{\mathsf{indisc}}\colon \mathsf{Sets}\to \mathsf{Cats}, \]
the indiscrete category functor, is the functor sending a set to its associated indiscrete category of Item 1.
