**Question.** Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback is always a weak equivalence). Or is there a close statement that I can't remember (browsing nLab did not help me) ?


**Comments.** Consider the diagram $\mathbf{D}=X\rightarrow Y \leftarrow Z$ where the left-hand map is a fibration and the right-hand map a weak equivalence. Let $T=\projlim \mathbf{D}$. Choose a fibrant replacement $Y^{fib}$ for $Y$ and a trivial cofibration $Y\rightarrow Y^{fib}$. Factor the composite map $X \rightarrow Y \rightarrow Y^{fib}$ as a composite trivial cofibration-fibration. We obtain a diagram $\mathbf{E}=X^{fib}\rightarrow Y^{fib}\leftarrow Z$ such that the left-hand map is a fibration between fibrant objects and the right-hand map is a weak equivalence. Let $U=\projlim \mathbf{E}$. By hypothesis, the map $U\rightarrow X^{fib}$ is a weak equivalence. By construction, the map of diagrams $\mathbf{D} \rightarrow \mathbf{E}$ is a weak equivalence of diagrams. The map $T\rightarrow X$ is a weak equivalence iff the map $T\rightarrow U$ is a weak equivalence. What next ?


**Why.** I found this cryptic remark in my notebook, and I can't remember where it comes from. The reason why I want to simplify the proof of right properness is that I have to deal with model categories where a set of generating trivial cofibrations is not known. I only know what I call a set of generating anodyne cofibrations. And the trivial fibrations which are the anodyne fibrations (i.e. having the RLP with respect to the set of generating anodyne cofibrations) which are a dual strong deformation retract. And the reason why I am interested in right properness is that I want to study right Bousfield localizations.



How to simplify the proof of right-properness?
To complete the argument you need to apply K. Brown's Lemma. Call your model category $\mathcal{M}$, then the map $Z \to Y$ induces a pullback functor $\mathcal{M} \downarrow Y \to \mathcal{M} \downarrow Z$ and the lemma implies that it preserves weak equivalences between fibrations over $Y$. If you define $V \to Y$ as the pullback of $X^{\mathrm{fib}} \to Y^{\mathrm{fib}}$, then $X \to V$ is a weak equivalence by the assumption and 2-out-of-3. Then apply the previous observation to $X \to V$ to see that $T \to U$ is a also weak equivalence and hence so is $T \to X$.


A reference for this is Lemma 9.4 in Bousfield's *On the Telescopic Homotopy Theory of Spaces*, but I imagine it was known well before that.


