Ag.algebraic geometry – Simplicial set represented by a set (unordered)

Let $ X $ to be a set (finished if you want) and form the whole simplicial $ F ^ { bullet} (X) $ with
$$
F ^ {n} (X) = mathrm {Hom} _ { mathrm {set}} ([n], X)
$$

where the right side indicates arbitrary maps of sets (of course
it would not be logical to say the order while preserving $ X $ do not come
with a command).

I wonder a description of $ F ^ { bullet} (X) $. For example if $ X = {0,1 } $ then there are 2 0-simplices, can also call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $ S ^ 1 $.

edit: As Goodwillie pointed out in his answer, this is not the end of the story, there are many more non-degenerate simplices of higher dimension.

Is there a similar description when $ X = {0, 1, 2 } $?

A closely related question is whether there is a deputy right to
forgetful functor of the simplex category $ Delta $ (ordered finished
sets), for example, finite sets (unordered) – and if so, what is it?

Example where such simplicial sets appear: given a map of the topological spaces $ f: X
to Y $
we can always form a
simplicial object $ mathcal {S} ^ { bullet} (f) $ with
$$
mathcal {S} ^ {n} = prod nolimits_ {X} ^ {n} = underlay {X times_ {Y}
cdots times_ {Y} X} _ {n text {times}}
$$

with facial maps and degeneracy given by projections and diagonals
respectively. Taking connected components gives a simplicial set.

When $ Y $ is the union $ bigcup_ {i = 1} ^ {N} H_ {i} $ of the coordinate
hyperplans in $ mathbb {C} ^ {N} $ and $ f: X = coprod_ {i = 1} ^ {N} H_ {i} to
bigcup_ {i = 1} ^ {N} H_ {i} = Y $
is the obvious map, I believe the simplicial
together we get is $ F ^ { bullet} ( {1, points, n }) $.