We can define a category $\mathcal{C}$ consisting of objects $(G,S)$ for a group $G$ and a subset $S\subseteq G$, and whose morphisms $(G,S)\to(H,T)$ are group homomorphisms $\varphi\colon G\to H$ such that $\varphi(S)\subseteq T$. Then the Cayley graph defines a functor $\Gamma\colon\mathcal{C}\to\mathrm{DirColGraph}, (G,S)\mapsto\Gamma(G,S)$ to the category of directed coloured graphs. Namely, given a map $\varphi\colon(G,S)\to(H,T)$ in $\mathcal{C}$, we have an associated map $\Gamma(G,S)\to\Gamma(H,T)$ that sends a vertex $g\in\Gamma(G,S)$ to the vertex $\varphi(g)\in\Gamma(H,T)$, that sends a colour $c_s$ to the colour $c_{\varphi(s)}$, and an edge $r_s(g)$ from $g$ to $gs$ of colour $c_s$ to the edge $r_{\varphi(s)}(\varphi(g))$ from $\varphi(g)$ to $\varphi(gs)=\varphi(g)\varphi(s)$ of colour $c_{\varphi(s)}$. It is clear that if $\varphi\colon(G,S)\to(G,S)$ is $\mathrm{id}_G$, then $\Gamma\varphi$ is also the identity map, and I will leave it to you to show that $\Gamma$ preserves composition of morphisms.
However, as category theorist I actually don't like directed graphs that much. Why not try to form categories? We can build a functor $\mathbb{E}\colon\mathrm{Grp}\to \mathrm{Grpd}$ towards the category of groupoids (i.e. categories in which each morphism is an isomorphism), sending a group $G$ to the groupoid $\mathbb{E}G$ whose objects are the elements of $G$, and in which a morphism $\varphi\colon g\to h$ is an element $s\in G$ such that $g\cdot s=h$. This means that, ignoring the colours, $\mathbb{E}G$ is an ''upgrade'' of $\Gamma(G,G)$ from a directed graph to a category (a groupoid, to be precise). There is another functor $\mathbb{B}\colon\mathrm{Grp}\to\mathrm{Grpd}$ that sends a group $G$ to a one-object groupoid $\mathbb{B}G$ with hom-set (singular) given by $\mathbb{B}G(\star,\star)=G$ (and composition is given by the group structure on $G$). There is a natural functor $\mathbb{E}G\to\mathbb{B}G$, which is of huge importance in algebraic topology, since it allows you to quickly construct Eilenberg-MacLane spaces and this functor is very important in the study of principal $G$-bundles.
If we want to introduce the subset $S\subseteq G$ again, we can define $\mathbb{E}(G,S)$ to be the subcategory of $\mathbb{E}G$ on all objects, in which a morphism $-\cdot h\colon g\to gh$ in $\mathbb{E}G$ lies in $\mathbb{E}(G,S)$ iff $h\in S$. This only works if $S$ is a subgroup of $G$! The reason is that our subcategory $\mathbb{E}(G,S)$ must be closed under composition. Note that the analogously defined groupoid $\mathbb{B}(G,S)$ will just be $\mathbb{B}S$, again under the assumption that $S$ is a subgroup of $G$.
If $S$ is a subgroup of $G$ and we want to have the colours again, we can work with the notion of a coloured category, in which each morphism has a certain colour. We simply colour the morphism $-\cdot s\colon g\to gs$ with the colour $c_s$, and obtain a functor $\mathbb{E}\colon\mathrm{Grp}^{(2)}\to\mathrm{ColGrpd}$, where $\mathrm{Grp}^{(2)}$ is the category of pairs $(G,S)$ of a group $G$ and a subgroup $S$ of $G$ (and whose morphisms are group homomorphisms preserving these subgroups), and $\mathrm{ColGrpd}$ is the category of coloured groupoids (and a morphism is a map of sets between the colours and a functor between the groupoids that acts accordingly on the colours).
