Individual set as category – is set a category? Is an individual set a category? I can imagine that each element can be considered an object and one can take as morphisms only those kinds of mappings among points that are composable associatively according to the axioms of category.
I have tried to google “set as category” (exact phrase) and there is no single Google answer that would say that an individual set can be a category indeed.
The consideration of the set as a category could be an interesting example or counterexample, that could clarify the meaning of a category.
If an individual set can be considered as a category, then more complex sets, e.g., individual manifolds, could be categories as well. Maybe there are some situations in which some benefits can be gained by having some more fine structure to the usual notions of the category of sets or category of manifolds.
 A: 
Is an individual set a category?

A category has three pieces of information: its objects, its morphisms, and its composition of morphisms.
A set, on the other hand, has only one piece of information, namely its elements.
So no, a set is not a category.
However, we may wonder if we can construct a category from a set.
As already mentioned in the comments, this is usually done by extending the set into a discrete category:

*

*Let $X$ be a set.
Let $\mathcal{D}$ be the following category:

*

*The objects of $\mathcal{D}$ are the elements of $X$.

*The only morphisms in $\mathcal{D}$ are the identity morphisms.

*The composition of morphisms in $\mathcal{D}$ is defined in the only possible way.

The category $\mathcal{D}$ is the discrete category associated to $X$.
(The term “discrete” refers to the fact that there exist no morphisms between distinct objects of $\mathcal{D}$.)
But this is not the only way of constructing a category from a set.
The following construction is arguably the second-most common one.


*Let $X$ be a set.
Let $\mathcal{I}$ be the following category:

*

*The objects of $\mathcal{I}$ are the elements of $X$.

*For every two elements $X$ and $Y$ of $\mathcal{I}$ there exists a unique morphism from $X$ to $Y$ in $\mathcal{I}$.

*The composition of morphisms in $\mathcal{I}$ is defined in the only possible way.

The category $\mathcal{I}$ is the indiscrete category associated to $X$.
These two constructions are the standard ways of associating a category to a set, and they are the standard ways for a good reason:
both constructions extend to functors
$$
 \mathrm{Disc}, \mathrm{Indisc} \colon \mathsf{Set} \longrightarrow \mathsf{Cat} \,,
$$
and together with the forgetful functor $\mathrm{Ob} \colon \mathsf{Cat} \to \mathsf{Set}$ these functors form a chain of adjunctions
$$
 \mathrm{Disc} ⊣ \mathrm{Ob} ⊣ \mathrm{Indisc}
$$
(i.e., $\mathrm{Disc}$ is left adjoint to $\mathrm{Ob}$ while $\mathrm{Indisc}$ is right adjoint to $\mathrm{Ob}$).
The functors $\mathrm{Disc}$ and $\mathrm{Indisc}$ are injective, full and faithful, whence they embed $\mathsf{Set}$ into $\mathsf{Cat}$ (in two different ways).
Because of this, people will often say that “sets are the same as discrete categories”.
(One could use the same logic to say that “sets are the same as indiscrete categories”, but that’s less common.)

If an individual set can be considered as a category, then more complex sets, e.g., individual manifolds, could be categories as well.

As already pointed out in the comments, we need to be careful here:
a manifold is not a special kind of set!
Instead, it consists of a set (of points) together with various extra structures on this set (a topology, a differentiable atlas, etc.).
Just as a set is not a category, a manifold is not a category either.
However, we may wonder which kinds of mathematical objects can be represented as certain kinds of categories.
For example, as alluded to above, sets, while not categories themselves, can still be represented as discrete categories.
The standard answers to this problem are

*

*sets and discrete categories,

*sets and indiscrete categories,

*monoids and one-object categories,

*groups and one-object groupoids,

*preordered sets and small thin categories,

*setoids and small thin groupoids.

All these examples are still fairly simple kinds of mathematical structures.
In some sense, this shouldn’t be too surprising:
trying to imitate mathematical structures with only objects, morphisms, and composition of morphisms will only take you so far.
If we want to represent more complex mathematical structures as categories, then we need to extend our definition of “category”.
As examples, we then get

*

*rings and one-object $ℤ$-linear categories,

*$$-algebras and one-object $$-linear categories, for any commutative ring $$,

*lots of examples I don’t know about.

