I am a bit confused on the notation.

>1. one specifies a set \$$\mathrm{Ob}(\mathcal{X})\$$, elements of which are called **objects**;

>2. for every two objects \$$x,y\$$, one specifies an element \$$\mathcal{X}(x,y)\$$ of \$$\mathcal{V}\$$.

According to the definition above, \$$x,y \in \mathrm{Ob}(\mathcal{X})\$$. But in order for \$$\mathcal{X}(x,y)\$$ to be an element of \$$\mathcal{V}\$$, \$$x,y \in V\$$ must be true also? Then what is the difference between \$$\mathrm{Ob}(\mathcal{X})\$$ and \$$V\$$?