In category theory, a dagger category is a category $\mathcal{C}$ such that for every pair of objects $A:\mathcal{C}$ and $B:\mathcal{C}$ there is a function $(-)^\dagger:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

• for all objects $A:\mathcal{C}$, $\mathrm{id}_A^\dagger = \mathrm{id}_A$,
• for all objects $A:\mathcal{C}$, $B:\mathcal{C}$, and $C:\mathcal{C}$, and morphisms $f:\mathrm{Mor}(A,B)$ and $g:\mathrm{Mor}(B,C)$, $(g \circ f)^\dagger = f^\dagger \circ g^\dagger$
• ${(f^\dagger)}^\dagger = f$.

In the context of higher category theory, does the concept of $(n,1)$-dagger categories make sense, and if so, how does one go about defining them?