# Universal Property of free objects

I am working on free objects, I am restricting myself primarely to groups, rings and modules (with maybe algebras) so in a sense in the concrete category (if I am not mistaken. This is a thesis work I am doing and as such references is of outmost importans, and I found this note document which gave an interesting definition of free objects which I'll cite

A free object is the mother of all objects of that type, in the sense that every object of that type is a quotient of the free object

This is an extremely good definition for what I am after, especially when you comapare it with this definition which I have seen many times before. I don't include the diagram or that one becuase I do not know how to draw a diagram on here. However I curious, does anyone know a good source for the first definition? Or how to prove it from the more common definition?

This is not a definition, it is a property of free objects. Every object surjecting onto a free object satisfies this too, and these do not have to be free. (Consider the group $S_n \times \mathbb{Z}$ for instance.) I advise you to read any text / book on universal algebra. There, you can find concise definitions and constructions of free objects.