Given that a Quiver (also known as directed multigraphs) is so important in category theory, being that every Quiver produces a free Category, and every Category has an underlying Quiver, I was wondering what a limit of a Quiver is. (See also the nice overview of Quivers and Diagrams in the answer on finding the limits of a diagram)
So if I take a category with two objects E and V and two functions s,t: E → V which given the start and end edges of vertices, then I should be able to have a functor from that to sets, which determines a quiver. Then all I need is to add a constant functor to some object of set say $L$ - the putative limit - and then find the real one. So for each vertex of the diagram, $v$, we have an arrow $p_v:L\to D(v)$. In this case it would be have the equations $p_V = s \circ p_E$, $p_V = t \circ p_E$. Which leads to $s \circ p_E = t \circ p_E$.
What would the co-version be? I suppose we turn the arrows around $p_v: D(v) \to L$ giving us $p_E = p_V \circ s$ and $p_E = p_V \circ t$, and so $p_V \circ s = p_V \circ t$
(I'd like to also ask what Monad one could build with the Free and Underlying Functors, but I guess that should be a different question).