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Assuming your universal property is true, it exactly says that the HNN extension is the coinserter of $(i_2 \circ \alpha,i_1) : H \to G$$(i_2 \circ \alpha,i_1) : H \rightrightarrows G$.
Assuming your universal property is true, it exactly says that the HNN extension is the coinserter of $(i_2 \circ \alpha,i_1) : H \to G$.
Assuming your universal property is true, it exactly says that the HNN extension is the coinserter of $(i_2 \circ \alpha,i_1) : H \rightrightarrows G$.
