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I was browsing through the questions and read one about whether defining a group as $G$ a set with certain features instead of an ordered pair $\langle G, \circ \rangle$, was abuse of language.

Someone mentioned that one could also define a group as an object of the Category of Groups. My question is: is that all one needs to say? are the group axioms implied from the category?

I haven't taken Category Theory so I apologize if the question is "stupid".

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    AFAIK, the "[category of groups](https://en.wikipedia.org/wiki/Category_of_groups)" is just the category the objects of which are those pairs and the morphisms of which are the usual functions. – 2017-02-08
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    If you're talking to someone who have never in their life seen a dog, you could try to explain what a dog is... Or you could just say that it is a member of the species "dog". That certainly explains everything. Your question is not stupid, but that "definition" is. – 2017-02-08
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    An aside: One can define a semigroup as a category with one object. (Semigroup elements are morphisms, multiplication is the composition of morphisms.) Of course, I would not advise to use this as a definition in an undergraduate abstract algebra class. – 2017-02-08

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