A (directed) graph $\Gamma$ is usually defined as a pair $(V,E)$ where $V$ is the set of vertices and $E \subseteq V \times V$ is the set of edges. A morphism of graphs $\Gamma$ and $\Gamma'$ is then a mapping $\varphi: V \to V'$ such that $(x,y) \in E$ implies $(\varphi(x), \varphi(y)) \in E'$.
I'm interested in thinking about inverse limits of (finite) graphs, ie. profinite graphs. Any time I see such a thing mentioned an alternative definition of a graph (due to Serre?) is usually used: A graph $\Gamma$ consists of a set $V$ of vertices, a set $E$ of edges and a mapping $\delta: E \to V \times V$, $x \mapsto (\delta_1(x), \delta_2(x))$. A morphism $\Gamma \to \Gamma'$ here is a pair of maps $\varphi_v: V \to V'$ and $\varphi_e: E \to E'$ such that $\delta_i(\varphi_e(e)) = \varphi_v(\delta_i(e))$ for all $e \in E$.
Is there any reason why we can't talk about an inverse system of graphs using the first definition and look at inverse limits of these? Perhaps something obvious breaks down that I'm just not seeing.