Set theory has a very convenient and well established curly brace notation to specify a set by its elements: $\{2,3,4,6\}$ or $\{\text{finite subgroups of }SU(2)\}$ are simple examples.
There should be a similar convenient notation for specifying a category by its objects and morphisms. Such a notation should easily accommodate categorical constructions such as slice categories. For example a double slash notation to separate objects and morphisms would define a slice category by something like (I am making this up!)
$$
\mathcal{C}\downarrow X= [ Y\to^f X : [Y//f]\in \mathcal C\quad // \quad (Y\to^f X)\to^h (Z\to^g X): [Y,Z//h] \in \mathcal C, g\circ h=f]
$$
(a commutative diagram in the second part of the specification would be more convenient here, but it should also be possible to typeset the notation inline).
Do such notations already exist? Whether they do or not, what notations would contributors recommend or suggest?
Update. Many thanks for comments made here. So far I most like the observation that clearly describing the morphisms makes the objects implicit. Still, I think beginners need the objects too, and have been experimenting with a notation like the one above, but using "staples" instead of square brackets, and introducing morphisms after objects by a vertical rectangular block (a bit like a closed staple). 2-morphisms could then be introduced in a similar way by a double block (a block with a vertical line through it). While the answers convince me that such notation is often unnecessary and maybe unhelpful sometimes, I'm not convinced such notation would be worthless.
Best Answer
One thing I often do is work with set-builder notation $\{blah\in thing| conditions\}$ where either or both of the $blah$ and $conditions$ are allowed to be (collections of) (2-)commuting diagrams. I also work with objects and arrows separately. (Edit: by which I mean I write $Obj(C) := \ldots$ and $Mor(C) := \ldots$ or similar)