Let $\mathscr C$ be a category.
Let $\text{Ob}(\mathscr C)$ be the set of all the objects of $\mathscr C$.
Is there a standard notation for $\bigcup_{A,B\in\text{Ob}(\mathscr C)}\text{Mor}(A,B)$?
And is $\text{Ob}(\mathscr C)$ the standard notation for the set of all the objects of $\mathscr C$?
I think usually people denote it by $Mor(\mathscr{C})$. You should be careful, because it is not always a set (if the category $\mathscr{C}$ is big) as Alex G. pointed out in his comment. But this union of all morphisms is always again a category with object being morphisms in $\mathscr{C}$ and maps between morphisms being the obvious commutative squares.