## Graph and Free Graph in Category Theory

Cat(FG, B) $$\cong$$Grph(G, UB)

Cat denotes the category of all small categories and Grph denotes the category of all small graphs.
G is a small graph which consists of small set O of objects and small set A of arrows f (CWM, PP48-51)
UB is a forgetful functor applied to category B which is an underlying graph of a category B.
Morphism of graphs D:G->UB corresponds to D$$\acute{}$$:FG->B

I can't figure out how D$$\acute{}$$ looks like and how the mapping behaves.
For D:G->UB, CWM (p50) says it sends each arrow f:a1->a2 of the given graph G to the string <a1,f,a2> of length 2 in UB.

Now, if G is freely generated to make a category FG, how is it generated and how does it look like?
A category itself can be described in a graph form. What would be the difference between B and UB if a forgetful functor is applied to B?

Any advice will be appreciated.

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