Category FunColimit

Let \(\mathcal{C}, \mathcal{D}\) be categories, then the category \(\mathrm{FunColimit}\) is the full subcategory of the functor category \(\mathrm{Fun}(\mathcal{C}, \mathcal{D})\) containing diagrams \(\mathcal{J}: \mathcal{D} \rightarrow \mathcal{D}\), which admit a colimit

\begin{align*} \mathrm{colim}(\mathcal{J}) \in \mathrm{Ob}(\mathcal{C}) \end{align*}