Preservation of Colimits. The functor $\pi _{0}$ of Item 1 preserves colimits. In particular, we have bijections of sets
\[ \begin{gathered} \begin{aligned} \pi _{0}\webleft (\mathcal{C}\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\mathcal{D}\webright ) & \cong \pi _{0}\webleft (\mathcal{C}\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}\pi _{0}\webleft (\mathcal{D}\webright ),\\ \pi _{0}\webleft (\mathcal{C}\mathbin {\textstyle \coprod _{\mathcal{E}}}\mathcal{D}\webright ) & \cong \pi _{0}\webleft (\mathcal{C}\webright )\mathbin {\textstyle \coprod _{\pi _{0}\webleft (\mathcal{E}\webright )}}\pi _{0}\webleft (\mathcal{D}\webright ), \end{aligned} \\ \pi _{0}\webleft (\text{CoEq}\webleft (\mathcal{C}\underset {G}{\overset {F}{\rightrightarrows }}\mathcal{D}\webright )\webright ) \cong \text{CoEq}\webleft (\pi _{0}\webleft (\mathcal{C}\webright )\underset {\pi _{0}\webleft (G\webright )}{\overset {\pi _{0}\webleft (F\webright )}{\rightrightarrows }}\pi _{0}\webleft (\mathcal{D}\webright )\webright ), \end{gathered} \]
natural in $\mathcal{C},\mathcal{D},\mathcal{E}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$.