simplicial stuff - What are the endofunctors on the simplex category?

To carry Charles' train of thought further:

By an 'interval' let us mean a finite ordered set with at least two elements; let $Int$ be the category of intervals, where a morphism is a monotone map preserving both endpoints.

The simplicial set $Delta^1$ can be viewed as a simplicial interval. That is, this functor $Delta^{op}rightarrow Set$ factors through the forgetful functor from $Int$ to $Set$. In fact, the resulting functor $Delta^{op}rightarrow Int$ is an equivalence of categories.

This extra structure (ordering and endpoints) on $Delta^1$ is inherited by Charles' $K_1=F^*Delta^1$; it, too, is a simplicial interval.

There aren't that many things that a simplicial interval can be. Its realization must be a compact polytope with a linear order relation that is closed. That makes it at most one-dimensional, and makes each component of it either a point or a closed interval. Simplicially each of these components can be either a $0$-simplex, or a $1$-simplex with its vertices ordered one way, or a $1$-simplex with its vertices ordered the other way, or two or more $1$-simplices each ordered one way or the other and stuck together end to end.

The three simplest things that a simplicial interval can be are: two points, a forward $Delta^1$, and a backward $Delta^1$. These arise as $F^*Delta^1$ for three examples of functors $F:Deltarightarrow Delta$, the only examples that satisfy $F([0])=[0]$, namely the constant functor $[0]$, the identity, and "op".

It's clear that any functor with $F([0])=[n]$ has the form $F_0coproddotscoprod F_n$ where $F_i[0]=[0]$ for each $i$. This means that the corresponding simplicial interval can be made by sticking together those which correspond to the $F_i$. For example, the 'shift' functor mentioned in the question is $idcoprod [0]$; Reid mentioned $idcoprod id$ and $opcoprod id$. These correspond respectively to: a $1$-simplex with an extra point on the right, two $1$-simplices end to end, and two $1$-simplices end to end one of which is backward. As another example, the constant functor $[n]$ corresponds to $n+1$ copies of (two points) stuck together end to end, or $n+2$ points.

In short, every functor $F:Deltarightarrow Delta$ is a concatenation of one or more copies of $[0]$, id, and op. I can more or less see how to prove this directly (without toposes or ordered compact polyhedra).