How to define the category of model structures of a category? It is possible to come up with different model structures for a fixed category. Let $\mbox{Models}\left(\mathcal{C}\right)$ be the category of all model structures of $\mathcal{C}$, which has as objects the triples of subcategories of $\mathcal{C}$  like $\left(W,F,C\right)$ satisfing the axioms of model structures and morphisms the appropriate endofunctors of $\mathcal{C}$. How can we clarify this appropriate. Ofcourse the trivial model structues are always there and it would be nice to define the morphisms of $\mbox{Models}\left(\mathcal{C}\right)$ in the way which they play a special role  (say universal objects). 
 A: The obvious categorical structure, in my view, is to put in a morphism $(C, W, F) \to (C', W', F')$ if and only if the identity functor is a right Quillen functor from $(C, W, F)$ to $(C', W', F')$, or more simply, if and only if $F \subseteq F'$ and $F \cap W \subseteq F' \cap W'$. Then:


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*The terminal object is the model structure where $C = \{ \text{isomorphisms} \}$ and $W = F = \{ \text{all morphisms} \}$.

*The initial object is the model structure where $C = W = \{ \text{all morphisms} \}$ and $F = \{ \text{isomorphisms} \}$.

*The model structure with $C = F = \{ \text{all morphisms} \}$ and $W = \{ \text{isomorphisms} \}$ doesn't seem to have a universal property in this category.

*This construction is compatible with dualisation, in the sense that $\mathrm{Models}(\mathcal{C}^\textrm{op}) = \mathrm{Models}(\mathcal{C})^\mathrm{op}$.


Admittedly this makes $\mathrm{Models}(\mathcal{C})$ into a preorder category, but then again, the set of topologies on a given set is more naturally a preorder category than anything else. However, unlike the set of topologies, this construction doesn't seem to be functorial.
