When answering this question,
I realized that I wasn't even sure what the correct definition of a model category was! When I first learned about the basics of model category theory, I learned from Mark Hovey's book "Model categories". Mark Hovey defines a model category to be
Definition 1.1.3. A model structure on a category C is three subcategories of C called weak equivalences, cofibrations, and fibrations, and two functorial factorizations (α, β) and (γ, δ) satisfying the following properties:
Hovey then goes onto describe the usual properties of a model category that we've all come to know and love. The interesting part here is that Hovey requires the factorization systems to be functorial. This means that one could have two different model structures on a category even if they have the same factorizing subcategories.
On the other hand, I have generally been told (and accepted) that model categories ought to be equipped with weak factorization systems, which does not require functoriality. I personally think this definition, which disagrees with Hovey's, is better suited for the underlying philosophy of homotopy theory, but I won't go into that.
Question: Is there a generally accepted concensus on what a model category is? Can anyone think of any propositions or theorems in Hovey's book, whose proofs rely heavily on functoriality of the factorization?