We claim that the functor
\[ F\colon \mathsf{Rel}^{\mathsf{op}}\to \mathsf{Rel} \]
given by the identity on objects and by $R\mapsto R^{\dagger }$ on morphisms is an isomorphism of categories.
By Chapter 9: Preorders, Item 1 of Proposition 9.6.8.1.3, it suffices to show that $F$ is bijective on objects (which is clear) and fully faithful. Indeed, the map
\[ \webleft (-\webright )^{\dagger }\colon \mathrm{Rel}\webleft (A,B\webright )\to \mathrm{Rel}\webleft (B,A\webright ) \]
defined by the assignment $R\mapsto R^{\dagger }$ is a bijection by Chapter 7: Constructions With Relations, Item 5 of Proposition 7.3.11.1.3, showing $F$ to be fully faithful.
We claim that the $2$-functor
\[ F\colon \mathsf{Rel}^{\mathsf{op}}\to \mathsf{Rel} \]
given by the identity on objects, by $R\mapsto R^{\dagger }$ on morphisms, and by preserving inclusions on $2$-morphisms via Chapter 7: Constructions With Relations, Item 1 of Proposition 7.3.11.1.3, is an isomorphism of categories.
By 
of , it suffices to show that $F$ is:
Thus $F$ is indeed a $2$-isomorphism of categories.
