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 $(-)^{\dagger_{A,B}}:\mathrm{Mor}(A,B) \to \mathrm{Mor}(B,A)$ such that

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

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?