In the spirit of this question, it would be interesting to give a characterization of HNN extensions as a 2-colimit. If $G$ is a group and $\alpha:H \xrightarrow{\cong} K$ is an isomorphism between two subgroups of $G$, then I think that the HNN extension $G_{\alpha}$ has the following universal property: if we let $i_1:H \hookrightarrow G,i_2:K \hookrightarrow G$ be the canonical inclusions, then the set of group homomorphisms $G_{\alpha} \to T$ are in natural bijection to pairs $(f,t)$ where $f:G \to T$ is a group homomorphism and $t:f \circ i_2 \circ \alpha \Rightarrow f \circ i_1$ is a 2-morphism. Here we're considering $\mathbf{Grp}$ as a full subcategory of $\mathbf{Cat}$ which carries a standard 2-category structure.

I'm just wondering if this universal property can be phrased as some kind of 2-colimit.