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Definition

Given a dagger 2-poset$A$, the category of maps$Map(A)$ is the sub-2-poset whose objects are the objects of $A$ and whose morphisms are the maps of $A$. In every dagger 2-poset, given two maps $f:hom_A(a,b)$ and $g:hom_A(a,b)$, if $f \leq g$, then $f = g$. This means that the sub-2-poset $Map(A)$ is a category and trivially a 2-poset.

Examples

For the dagger 2-poset of sets and relations $Rel$, the category of maps $Map(Rel)$ is equivalent to the category of sets and functions $Set$.