In graph theory a pseudograph is a particular type of graph. A pseudograph is a set of vertices and for each unordered pair of vertices a set of edges between these.

where Pseudograph is the category of pseudographs and DagCat? is the category of (smallstrict) dagger categories. Moreover, this forgetful functor has a left adjoint

is an involution$\dagger \circ \dagger = \mathrm{id}_G$.

Nuts and bolts definition

A pseudograph is a quiver$(E, V, s, t)$ with a function $(-)^\dagger:E \to E$ such that for any edge $f:E$, $s(f) = t(f^\dagger)$, $t(f) = s(f^\dagger)$, and $(f^\dagger)^\dagger = f$.