Category of Grp is not a subcategory of Set I am trying to read the book Rings and categories of modules by Anderson and Fuller.
On page 7 it is stated that the category of Groups is not a subcategory of a category of Sets, some explanations are given there. Can any one please give me some more details about? especially why $mor_G((G,\circ),(H,\circ))\not\subset Map((G,\circ),(H,\circ))$ ?
 A: Well, if one wishes to be pedantic (as the cited work is), then one notes that $(G, \circ)$ and $(H, \circ)$ are particular one- or two-element sets (if we use the Kuratowski definition of ordered pair), so the set of maps $(G, \circ) \to (H, \circ)$ has at most four elements. Of course, there could be more than four homomorphisms $(G, \circ) \to (H, \circ)$.
You should not be overly concerned with such fine distinctions.
A: Your interpretation of the text on page 7 is not right (which is, I admit, a bit confusing).
A group $G = (|G|, {\circ})$ is a pair consisting of a set and an operation (satisfying certain axioms).  Thus, a group is not a set.  It is a structure made up of a set along with other stuff.
It is common mathematical practice to use the same symbol for the group as well as its underlying set and write $G = (G, {\circ})$.  The two occurrences of $G$ in this formula denote different things, one a group and the other a set.  It would be clearer to write what I have written above using different symbols for the two.  But normal mathematicians are expected to recognize that two things are being talked about, and figure out which of them is meant whenever one sees the symbol "$G$".  When we write "$h : G \to H$," we are referring to groups $G$ and $H$.  When we write "$x \in G$," we are referring to the underlying set of $G$, i.e., $|G|$ in my notation.
Grp is not a subcategory of Set merely because their collections of objects are different kind of things.
(You can ignore Anderson & Fuller's justification which is saying the same thing, but in a more complicated way perhaps.)
