Homotopy coherent companionships and conjunctions

We demonstrate that companionships and conjunctions in double $\mathrm{\infty }$-categories – and more generally, in double Segal spaces – extend to functors out of the free-living companionship and conjunction respectively. Specifically, we prove that these extensions are (homotopically) unique: the corresponding spaces of extensions are contractible under suitable completeness assumptions. The developed theory is then put to use to give a characterization of companions and conjoints in functor double Segal spaces in terms of so-called companionable and conjointable 2-cells. We end with an application of our results to $(\mathrm{\infty },2)$-category theory.