Quivers and quiver homomorphisms form a functor category of set-valued diagrams

Let $\mathcal{J}$ have objects $E,V$, identities, and two distinct arrows $s,t:E\to V$. A functor $Q:\mathcal{J}\to \mathbf{S}\mathbf{e}\mathbf{t}$ consists of a set $Q(E)$ of edges, a set $Q(V)$ of vertices, and source and target maps $Q(s),Q(t):Q(E)\to Q(V)$. This is exactly a quiver.

A natural transformation $\alpha :Q\Rightarrow Q^{\mathrm{\prime }}$ consists of functions ${\alpha }_E:Q(E)\to Q^{\mathrm{\prime }}(E)$ and ${\alpha }_V:Q(V)\to Q^{\mathrm{\prime }}(V)$ with ${\alpha }_{V}Q(s)=Q^{\mathrm{\prime }}(s){\alpha }_E$ and ${\alpha }_{V}Q(t)=Q^{\mathrm{\prime }}(t){\alpha }_E$. These are precisely the incidence-preservation equations for a quiver homomorphism.

Since identities and composition are componentwise in the functor category, this identification respects identity quiver maps and their composites. Hence quivers and quiver homomorphisms form $[\mathcal{J},\mathbf{S}\mathbf{e}\mathbf{t}]$.