Define a model structure on $\bf Cat$ by the following rules:
- A weak equivalence is an equivalence of categories;
- A cofibration is a functor which is injective on objects;
- A fibration is a functor $F\colon \bf C\to D$ such that for all $C\in \bf C$, for all isomorphism $f\colon F(C)\cong D$, there exists an object $C'\in \bf C$ and an isomorphism $f'\colon C\cong C'$ such that $F(f')=f$, $F(C')=D$.
I'm stuck in proving that these condition really define a model structure on $\bf Cat$, and in particular I'm not able to show that in a diagram $ \begin{array}{ccc} \bf C &\xrightarrow{U}& \bf K \\ F\downarrow&&\downarrow G \\ \bf D &\xrightarrow[V]{}& \bf L \end{array} $
(LLP) if $G$ is an acyclic fibration and $F$ a cofibration, then there exists a filling arrow $W\colon \bf D\to K$ making the diagram commute.
(RLP) if $G$ is a fibration and $F$ an acyclic cofibration, then there exists a filling arrow $W\colon \bf D\to K$ making the diagram commute.