The definition that appeared in Francesco's answer is in the same spirit as ($\infty,1$)-adjoint pairs.

Dear Mathieu, I think what Martin proposes is not necessarily Puppe-exact. For example, the category of pointed sets is not Puppe-exact. The epimorphisms are not cokernels in general.

I want to comment that the pullback of a right horn along a left fibration is a right anodyne map. One may find this statement in Rezk's lecture notes ' Stuff about quasicategories'. But I think he/she does not provide a proof.

I am confused by your last sentence. Given two homotopic maps from $K\times \Delta^1$ to $X$, is it clear how to prove the induced maps from $K\times \Delta^1$ to $X_{s'}$ are homotopic?

I mean the reference is still not correct. I find a probably right reference which uses set theory and no countability is needed. The reference is Bourbaki Theorie des Ensembles, page 162, Theorem 1. Anyway, I'll check the details.

That's the point, my version says that the index set has a numerable subset. Can you show me the quoted result directly? Mine is: Theorem 1 (Mittag-Leffler)- Let (X_{\alpha,f_{\alpha,\beta}}) be a projective system of uniform, complete and separated spaces, indexed by a filtered set $I$ which admits a numerable cofinal subset; we suppose in addition that for any $\alpha\in I$, $X_{\alpha}$ possesses a numerable fundamental system of entourages.

Am I missing something? The result in Bourbaki's book requires a numerable subset, however, I can't see why this is true in his arguments.

Note that the definition of $\omega-$smallness of Hovey might not be what we want. Let the cardinal of $\lambda$ be $\aleph_0$. Then $\lambda$ is not $\lambda-$filtered in the sense of Hovey. But if the set of maps contains all identities, one may extend a smaller sequence to a larger sequence obviously.

Also in Thm 1.2.3.5 (4) he uses the smallness which by the above argument is equivalent to compactness since the ordinal is $\omega$. See mathoverflow.net/questions/188714/…

First we have a cofinal functor from a directed category to a given filtered category. Then a directed category is a union of a $\lambda-$sequence of directed sub categories (whose cardinalities are strictly less than that of the directed category). A colimit of a $\lambda-$sequence of fibrations is a fibration by Hovey's proof. So we are reduced to prove that for each directed sub-category, the directed colimit of fibrations is a fibration. To prove this, we can use induction on the cardinality of the directed category.

If the forgetful functors for the algebras $S$ and $T$ are fully faithful and $F^*$ is an equivalence of categories, can we deduce that $F$ is also an equivalence of categories?