Is there a natural homotopy inverse to the map $∣Sing(X)∣\rightarrow X$

Question

Let $X$ be an object of the category $C$ of CW complexes.

Let $Sing:C \rightarrow SSet$ be the functor sending a CW complex to its singular complex (a simplicial set).

Let $∣-∣: SSet \rightarrow C$ be the geometric realization functor.

We have a canonical map that is a homotopy equivalence $\epsilon_X: ∣Sing(X)∣\rightarrow X$.

Is there a natural homotopy inverse? Said differently, is there a natural transformation $\text {Id} \Rightarrow∣Sing(-)∣$ consisting of homotopy inverses to the $\epsilon_X$’s?