Suppose I have a category $\mathbf{C}$ and classes of morphisms $\mathcal{W}$ and $\mathcal{C}$, and I would like to know that $\mathcal{W}$ and $\mathcal{C}$ are the weak equivalences and the cofibrations of a model category structure.

I can certainly write down what the fibrations $\mathcal{F}$ would have to be, and I'm wondering if there are any theorems to provide shortcuts in verifying that the classes $\mathcal{W}$, $\mathcal{C}$ and $\mathcal{F}$ actually satisfy all the conditions of a model category.