Is this functor a left adjoint?

Question

I have a functor $sGrph \to Quiv$ which replaces each edge in a given simple graph with a cospan (and leaves the vertices alone). This functor clearly preserves colimits. Is it a left adjoint?

Answer 1

Let's call your functor $F.$ Let $2$ denote the two-object one-arrow quiver that looks like $0\to 1.$ Then $|\mathrm{Hom_{Quiv}}(F(K_2),2)|=1,$ but $|\mathrm{Hom_{sGraph}}(K_2,G)|$ is even for any simple graph $G.$ So $F$ cannot be a left adjoint.

To use an adjoint functor theorem to construct a right adjoint you'd really want $\mathrm{sGraph}$ to be cocomplete, but it doesn't have a coequalizer for the two automorphisms of $K_2$ for example.