#this shows that the image of the diagonal functor from Set to Q^hat is both reflective and coreflective

[bubbloquacious]

Why yes I am still thinking about graphs as presheaves. Another thing you can do with functors from small categories is think about their limits and colimits. We usually think of limits and colimits as objects of the codomain category, so in this case the limit of a graph-as-functor would be a set. However, we can turn it back into a graph by then applying the diagonal functor Δ from the category of sets to the category of graphs.

Let's think about what this functor does. It maps a set X onto the constant functor from Q^op at X. Such a functor maps every object of Q onto X and every morphism of Q onto the identity function of X. So, the vertex set and the arrow set of Δ(X) are equal, and the source and target functions are the identity. The graph we get consists of one vertex for every element of X and each vertex has a single loop.

A cone under or cocone over a functor F: C -> D is exactly a natural transformation from a constant functor to F or from F to a constant functor, respectively. The limit of F is the terminal object in the category of cones under F, so in addition to the limit object we have a natural transformation from the constant functor at that object to F. Dually for colimits. Natural transformations in Q^hat are graph homomorphisms, so we can do some fun interpretation here.

The limit of a functor with domain Q^op (or Q, as they are isomorphic) is an equalizer. In Set, the equalizer of functions f,g: X -> Y is the set { x ∈ X : f(x) = g(x) } (up to a canonical bijection). It follows that the limit of a graph is exactly the set of arrows of G whose source vertex coincides with their target vertex. The set of all loops of G. Applying Δ to this we get the graph that has a vertex for every loop, and a loop on each of these vertices. The universal cone from this graph to G is the graph homomorphism that maps each loop in this graph onto the corresponding loop in G.

The colimit of a functor from Q^op is a coequalizer (surprise surprise), and in Set the coequalizer of f and g is the quotient set of Y by the smallest equivalence relation ~ such that f(x) ~ g(x) for all x ∈ X. In a graph G we have that Y is the vertex set, X is the arrow set, and f and g are the source and target functions. This means that if two vertices v and w are connected by some arrow, then they will be equivalent under ~. Applying transitivity and symmetry, we find that the equivalence classes of ~ are exactly the connected components of G. The universal cocone over G is the graph homomorphism that maps all vertices in a connected component of G onto the corresponding vertex of the graph-that-has-a-vertex-for-every-connected-component, and all arrows in that connected component onto the unique loop on that vertex.