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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.

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I think you are looking for the definition of a full subcategory. This is defined as follows:

Let $S$ be a subcategory of a category $C$. We say that $S$ is a full subcategory of $C$ if for each pair of objects $X$ and $Y$ of $S$ we have $$ \operatorname{Hom}{}_S(X,Y) = \operatorname{Hom}{}_C(X,Y) $$

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).

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