I am looking at the category of graphs. I am learning that if we have a morphism between 2 graphs (where the graphs come via an insertion from the endomaps of sets), then the map itself comes from the insertion from a map in the category of endomaps. They call this fullness. Is it possible to give an example that explains this more clearly? I am not clear on what this means.
I think you are looking for the definition of a full subcategory. This is defined as follows:
In other words, a subcategory $S \subset C$ is full if the morphisms of $S$ coincide with the ones from $C$.
In your case, I guess the category of graphs is a full subcategory of another category (the one of endomaps of sets maybe? I don't really know what this means).