Sets

It is the most important category, the inspiration for the others, maybe.

• Objects: the collection of all sets
• Morphisms: maps between sets
• Composition is, of course, the composition of maps.

It has the structure of a lattice.

Here’s a version of your note with those additions integrated:

Membership as a morphism

In Set, the usual notion of membership can be expressed categorically:

• An element $x\in A$ is the same thing as a morphism

where $\{\ast \}$ is a singleton set.
This way, an element is identified with a function from the “one-point set” into $A$, sending $\ast \mapsto x$. This interpretation is often called the idea of a generalized element.

Generalized elements in other categories

The notion extends beyond Set:

• In any category $\mathcal{C}$ with a terminal object $1$, a morphism

plays the role of an element of $A$.

• This provides a generalized notion of belongingness or membership, even if objects of $\mathcal{C}$ are not sets in the usual sense.

Thus, in Set, the terminal object is the singleton set, and we recover the familiar definition of an element. In other categories, these morphisms encode “points” or “states” of objects. This is related to Yoneda’s lemma, which is where generalized elements become formalized as morphisms from arbitrary objects, not just the terminal one.
For an interesting example see random variable#Category theory viewpoint.