Adjointness. We have an adjunction witnessed by a bijection of sets
\[ \textup{Hom}_{\mathsf{Grpd}}\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\mathcal{G}\webright )\cong \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{G}\webright ), \]
natural in $\mathcal{C}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and $\mathcal{G}\in \text{Obj}\webleft (\mathsf{Grpd}\webright )$, forming, together with the functor $\mathsf{Core}$ of Item 1 of Proposition 8.3.3.1.4, a triple adjunction
witnessed by bijections of sets
\begin{align*} \textup{Hom}_{\mathsf{Grpd}}\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\mathcal{G}\webright ) & \cong \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{C},\mathcal{G}\webright ),\\ \textup{Hom}_{\mathsf{Cats}}\webleft (\mathcal{G},\mathcal{D}\webright ) & \cong \textup{Hom}_{\mathsf{Grpd}}\webleft (\mathcal{G},\mathsf{Core}\webleft (\mathcal{D}\webright )\webright ),\end{align*}
natural in $\mathcal{C},\mathcal{D}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and $\mathcal{G}\in \text{Obj}\webleft (\mathsf{Grpd}\webright )$.