Finite directed paths form the free category on a quiver

A path of length $n$ is a composable $n$-tuple of edges. Concatenation of paths is associative, and the length-zero path at a vertex is a two-sided identity. Thus these data form a category $P(Q)$ by [F1], including when $Q_0$ is empty.

Given a quiver map $q:Q\to U(\mathcal{C})$, define $\widehat{q}:P(Q)\to \mathcal{C}$ on vertices by $q$ and on a path by composing the images of its edges in order; send an empty path to the appropriate identity. Associativity in [F1] makes this a functor.

For a quiver map $f=(f_0,f_1):Q\to Q^{\mathrm{\prime }}$ define $P(f):P(Q)\to P(Q^{\mathrm{\prime }})$ by $f_0$ on objects and by $(e_1,\dots ,e_n)\mapsto (f_1{e}_1,\dots ,f_1{e}_n)$ on paths; the image tuple is composable because $f$ commutes with source and target. It sends empty paths to empty paths and concatenations to concatenations, so it is a functor, and $P(1_Q)=1_{P(Q)}$ and $P(gf)=P(g)P(f)$ hold componentwise. Thus $P$ is a functor.

Any functor extending $q$ must have the values in step 2.1: induction on path length using [F2] forces the empty path to an identity and each longer path to the composite of its edge images. Hence the extension is unique.

Both $\mathbf{Q}\mathbf{u}\mathbf{i}\mathbf{v}$ and $\mathbf{C}\mathbf{a}\mathbf{t}$ here consist of small quivers and small categories, so a quiver map is a pair of functions between sets and a functor is a pair of functions between sets; each morphism collection is a subset of a set of functions and hence a set, making both categories locally small as [L1] requires. Restriction to vertices and edges and the extension $q\mapsto \widehat{q}$ are inverse by steps 2.1 and 3.1, and they commute with the actions of step 2.2 and with postcomposition by functors, so the bijection $\mathbf{C}\mathbf{a}\mathbf{t}(P(Q),\mathcal{C})\cong \mathbf{Q}\mathbf{u}\mathbf{i}\mathbf{v}(Q,U(\mathcal{C}))$ is natural and [L1] yields $P⊣U$.