In the spirit of [this question](https://mathoverflow.net/questions/96078/are-semi-direct-products-categorical-colimits), 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.



HNN-extension as a 2-colimit
Assuming your universal property is true, it exactly says that the HNN extension is the [coinserter](https://ncatlab.org/nlab/show/inserter) of $(i\_2 \circ \alpha,i\_1) : H \rightrightarrows G$.


