I have a question about a statement in Mac Lane's "Categories for the working mathematician". It is in page 51, in the context of graphs and free categories. The statement basically says:
Let $G$ be a graph, $C_G$ its associated free category and $p: G \to UC_G$ the canonical map where $U: \operatorname{Cat} \to \operatorname{Grph}$ is the forgetful functor. For any category $B$, we have a natural bijection (by the universal property of $C_G$) $\operatorname{Cat}(C_G,B) \cong \operatorname{Grph}(G,UB)$ given by $F: C_G \to B \mapsto UF \circ p$. (Here comes the part I have trouble with) Moreover, this bijection is natural in G and in B.
What I am not understanding is what does this last sentence mean in this context?
I tried to construct the most natural functors possible out of the maps $B \mapsto \operatorname{Cat}(C_G,B)$ and $B \mapsto \operatorname{Grph}(G,UB)$, which led, for example, to the functor $\operatorname{Grph}(G,U \bullet): \operatorname{Cat} \to \operatorname{Cat}$, by defining a category structure in $\operatorname{Grph}(G,UB)$, (since it doesn't seem to have an intrinsic one, since $G$ and $UB$ also don't) by taking as objects the morphisms of graphs $A: G \to UB$ and as arrows the transformations $\tau: A \to A' $ defined by $\tau_c:A(c) \to A'(c)$ if such arrows exist in $UB$. After this, $\operatorname{Grph}(G,UB)$ has only a graph structure, so I took the final structure to be the free category associated with that graph... I tried to follow this approach for a bit to see if I could get some natural functors out of those classes and then construct a natural transformation between the two functors from the bijections, but this seems rather farfetched to me, and seems to have a lot of problems, and so I couldn't get a conclusion out of this.
Is my interpretation wrong?
I appreciate any help and thank you in advance :)
Let $A,A' : G \to U(B)$ be two graph morphisms (which you can also see as diagrams of shape $G$ in $B$). A morphism $A \to A'$ is by definition a family of morphisms $A(v) \to A'(v)$, for each vertex $v$ of $G$, such that for each edge $v \to w$ in $G$ the diagram $$\begin{array}{cc} A(v) & \rightarrow & A'(v) \\ \downarrow && \downarrow \\ A(w) & \rightarrow & A'(w) \end{array}$$ of morphisms in $B$ commutes. You can easily construct compositions of such morphisms and show that $\mathrm{Graph}(G,U(B))$ is, indeed, a category. Notice the similarity to the notion of a natural transformation. This can be used to show that the bijection
$$\mathrm{Cat}(C_G,B) \to \mathrm{Graph}(G,U(B)),~ F \mapsto F|_G$$
extends to an isomorphism of categories. The verification of naturality is trivial.