Let $(\mathcal{M}, \mathcal{W}, \mathcal{C},\mathcal{F})$ be a model category and $\text{Ho}(\mathcal{M})$ its homotopy category. Now we consider $ \mathcal{M}_{\mathcal{F}}$ the full subcategory of fibrant object. Now, I want to show that we have an equivalence between $\text{Ho}(\mathcal{M})$ and $\text{Ho}(\mathcal{M}_{\mathcal{F}})$.
For that I first construct a functor: $$\mathcal{M}_{\mathcal{F}} \xrightarrow{\subset} \mathcal{M}\xrightarrow{\lambda}\text{Ho}(\mathcal{M})$$ Where $\lambda$ is the localization functor. Then since by definition, this functor sends weak equivalences to isomorphism, we get a functor $\text{Ho}(\mathcal{M}_{\mathcal{F}})\to \text{Ho}(\mathcal{M})$.
Now I want to construct a functor $text{Ho}(\mathcal{M}\to \text{Ho}(\mathcal{M}_{\mathcal{F}}))$: For that I start to construct the following functor: $$R: \mathcal{M}\rightarrow \mathcal{M}_{\mathcal{F}} $$ We define $R$ on object: We take $X\in \text{Ob}(\mathcal{M})$, and consider the unique morphism $X\to *$, where $*$ is the final object of $\mathcal{M}$, by the factorization axiom we get : $$X\overset{\sim}{\hookrightarrow}R(X) \twoheadrightarrow *$$ We take $R(X)$ to be the object in the middle. Now I want to define $R$ on morphism, and in my definition of a model category, I didn't add the fact that this decomposition should be functorial, so if I consider a morphism $f:X\rightarrow Y$ in $\mathcal{M}$ I don't know how to construct a morphism $R(f): R(X)\rightarrow R(Y)$ such that $R$ is indeed a functor. Any hints on what I should use to construct such a morphism? And more importantly am I on the right track to prove my point ?
I don't think there is a substantial difference in complexity between showing full and faithfulness and showing an inverse functor exists.
Note that a morphism $f:X\rightarrow Y$ gives rise to a commutative square of the form $$\begin{array}{ccc} X & \overset{f}{\longrightarrow} Y \longrightarrow & RY\\ \downarrow^\text{triv}_\text{cofib} &&\downarrow_\text{fib}\\ RX &-\!-\!-\!-\!\longrightarrow&\ast \end{array}$$ so you get a lift $RX \rightarrow RY$. If you show that this lift is unique up to homotopy, you can use that either to define the inverse functor on morphisms, or deduce from it that the inclusion is faithful (it is obviously essentially surjective and full).