TITLE: How canonical is cofibrant replacement?
QUESTION [24 upvotes]: Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration.  More recent references have strengthened this axiom by assuming that this is actually a functorial factorization.
If you take a "classical" model category (not exactly in Quillen's sense - I would still like to assume that it has all small limits and colimits), it becomes natural to ask whether it has such a functorial factorization.  More, one might wonder how canonical this factorization is, even just on the level of objects.
So in this situation, let's say that we have either:

a very large category whose objects are "factorization functors", and whose morphisms are natural transformations between them (of necessity natural weak equivalences), or
a very large category whose objects are functorial cofibrant replacements for objects, and whose morphisms are again natural weak equivalences.

Are there easy examples where these categories are empty?  Are there examples where they are nonempty, but the functorial factorization is "noncanonical" in the sense that the category of factorizations is noncontractible?  Does this category of factorizations become contractible under stronger assumptions (such as cofibrant generation)?

Added later: It turns out that the question of homotopy type has a boring answer (and, embarassingly, one answerable by the standard techniques).  Let's suppose we have one factorization functor $F$, so that any arrow $g:x \to y$ factors canonically as $$x \stackrel{c(g)}{\to} F(g) \stackrel{f(g)}{\to} y.$$
Then, naturally associated to any other such functor $G$, we get a third replacement functor abusively written as $G \circ F$, obtained by applying $G$ to the map $c(g)$; this gives a factorization $$x \to (G\circ F)(g) \to F(g) \to y$$ and the first map, together with the composite of the latter two maps, gives a new factorization.
There are natural transformations of factorization functors $F \leftarrow G \circ F \rightarrow G$, and this is natural in $G$; this provides a two-step homotopy contracting the space of factorization functors down to the constant $F$.
(However, the responses to the first question have already informed me quite a bit!)

REPLY [10 votes]: Dear Tyler,
The other replies here have summarised quite well what has been done by Emily and myself in this regard, but it might be useful to point something out concerning the later addition you have made to your question. You describe a construction that, from two functorial factorisations $F$ and $G$, produces a third, $G \circ F$. This, as you observe, comes equipped with a natural transformation $G \circ F \to F$. However, in trying to define similarly a natural transformation $G \circ F \to G$, one must use lifting properties; the component at $g$ being obtained by considering a square with $x \to G \circ F(g)$ on its left, and $G(g) \to y$ on its right. The problem is that these components are highly unlikely to constitute a natural transformation; the naturality squares most likely do not commute. Even if the functorial factorisations in question were derived from Quillen's small object argument, and the liftings chosen were the ones described in that argument, naturality is still unlikely to obtain: for the construction of those liftings requires the making of some non-canonical choices (more specifically, the choices made in Hovey's book, p.33, line 6, "there is a $\beta < \gamma$") which materially affect the resulting fillers, and are unlikely to cohere for different choices of $g$. 
The notion of "natural" / "algebraic" weak factorisation system could be understood as an attempt to rectify precisely this problem. Here one has provided not only the functorial factorisation, but in addition, canonical choices of filler, which, in particular, will cohere sufficiently to allow the construction of the natural $G \circ F \to G$ sought above. That every (cofibrantly generated) weak factorisation system can be made into an instance of this notion relies on modifying the small object argument in such a way as the chosen fillers it provides no longer rely on the making of non-canonical choices (to be more precise: the non-canonical choices are still there, but the outcome is now independent of them).
Richard