I've been learning about model category theory and cofibrant generation of model structures, and I've come across Quillen's small object argument. In the informal paper I'm writing I'm choosing to omit the proof of the small object argument, but instead I wanted to show an example of explicitly using the small object argument to construct a factorisation of a morphism but I don't really know where to start or how to do it?
I'd like to apply it to a simple morphism in $\textbf{Cat}$, the category of small categories, since I've used it as an example to demonstrate several other things in the paper. Perhaps to the functor $\textbf{1} \rightarrow \textbf{2}$, where $\textbf1$ is the one object category and $\textbf2$ is the two object category with one non-trivial isomorphism.
Any advice on how to approach this problem would be very much appreciated!
edit: I'm using the canonical model structure:
W is the class of category equivalences;
Cof is the class of functors that inject on objects;
Fib is the class of isofibrations: link