My definition of 2-UGraph (undirected graphs with at most one edge from any given vertex to another) as "the full subcategory of Set-UGraph (= Set^M^op for M the monoid Set(2,2)) induced by the evident functor Nonempty:Set->2 collapsing nonempty homsets to singletons" was an attempt to say "2-UGraph is a retract of Set-UGraph" with both too few words and too many. 2-UGraph is the full subcategory of Set-UGraph consisting of graphs with at most one edge per "homset", and at the same time the quotient of Set-UGraph arising from identifying all members of each homset of each graph. I.e. a retract. This is something of an eye-opener for me as I have for decades thought of the undirected graph (of the one-edge-per-homset kind) as the algebraically impoverished cousin of the directed graph. I am tickled pink to find it arising as a retract of a presheaf category, and moreover without either of the two quirks that have been pointed out for the more general undirected graphs allowing multiple edges per homset (Set-UGraph has two types of distinguished loop, and does not embed in Set-DGraph). Vaughan Pratt