Let $X$ be a based space. Then the Moore loop space $MX$ is defined to be the topological monoid whose points are based loops $[0,a] \to X$ where $a \ge 0$ is allowed to vary. Composition is gotten by concatenating loops.

Since $MX$ is a topological monoid, we can form the *bar construction* $BMX$.
This is geometric realization of the simplicial space
$$
[n] \mapsto (MX)^{\times n}
$$
where the face and degeneracy maps are defined using: projection, multiplication, and
insertion of the identity element.

If $X$ is a connected based CW complex, then there is a homotopy equivalence $$ BMX \simeq X . $$ A standard way to prove this is to construct a quasi-fibration $EMX \to BMX$ whose fiber at the basepoint is identified with $MX$ up to homotopy equivalence, in which $EMX$ is contractible. The quasi-fibration is functorially associated with $MX$. However, it seems to me that the equivalence $BMX \simeq X$ gotten in this fashion depends on a contractible space of choices. In particular it doesn't seem to be natural.

**Question:**

Is there a zig-zag of *natural transformations*
$$
BMX = f_0(X) \leftarrow f_1(X) \to f_2(X) \leftarrow \cdots \to f_n(X) = X
$$
which yields a chain of equivalences when $X$ is a connected CW complex?