sSet-enriched categories, quasi-categories and the model-independent theory sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual examples of $(\infty,1)$-categories.
In fact, to give actual examples of $\infty$-categories in the model of quasi-categories, one uses various kind of nerves to get quasi-categories from various types of enriched categories, among which sSet-enriched are a notable example.
For the model given by quasi-categories, also called $\infty$-categories following Lurie's terminology, there are very good references which develop the common 1-categorical notions in the higher setting, for eg. Kerodon, HTT, or the book by Cisinski. I haven't found a similar treatment for sSet-enriched categories.

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*The approach of Riehl and Verity (see for eg. this and this) is particularly appealing to me, but if I understand correctly their synthetic theory does not apply to the model given by sSet-enriched categories, is this correct or am I misunderstanding?

*Is there a reference which develops the basic theory for the model given by sSet-enriched categories?

*For example, what is an adjunction between $(\infty,1)$-categories
presented by sSet-enriched categories? The naive guess would be that
it is just a sSet-enriched adjunction? What is an equivalence?

*If the synthetic theory of Riehl and Verity does not apply to the
model given by sSet-enriched categories, are we in the situation
where we have a theory well developed for some models, but some
important actual examples can only be given by applying some functor,
e.g. the homotopy coherent nerve to get quasi-categories from
sSet-enriched categories?

 A: This has not been done, and there are good reasons for it: While $sSet$-enriched categories are indeed very good to easily get examples of $\infty$-categories, they are very bad at understanding what are functors between $\infty$-categories. The problem is that a "sSet-enriched functors" is a way too strict notion, that need to satisfies a very strong condition of compatibility with composition, while the correct definition of functor between $\infty$-categories should only satisfy this condition up to equivalence.
When you work with quasi-categories or Complete Segal spaces, $\infty$-functors are justs the morphisms of simplicial sets or bisimplicial sets.
For example, to answer your third question: sSet-enriched adjunction definitely induce adjunction between the associated $\infty$-categories, but not all such adjunction will be of this form (for fixed simplicially enriched categories).
The reason while we can use sSet-enriched categories to model $(\infty,1)$-categories despite this problem is because there are enough "nice" sSet-enriched category that have the properties that any "$\infty$-functors" (or "weak functor", whatever this should mean) out of them is equivalent to one that is a sSet-enriched functor (this is a kind of stratification result if you'd like). These "nice" sSet-enriched categories are essentially the cofibrant objects of the Bergner model structure. And "enough" mean that one can always take cofibrant replacement.
So if one wanted to get some sort of nice "synthetic" theory that applies to sSet-enriched categories, or just a setting where one could develop $(\infty,1)$-category theory nicely, then one would need to either:

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*Move to a category of sSet-enriched categories and "weak functors" between them (for some definition of weak functor), but this is a fairly complicated category, with poor category theoretic properties. So not a very nice place to work in. It is much simpler to consider that a good way to define these weak functors is as morphisms between the associated quasi-categories, or between some associated complete Segal spaces, and works in these categories instead.


*Keep taking "cofibrant replacement" all the time everywhere in the theory. This is surely possible, but it makes everything much more complicated and given that we have model where this is not needed (quasi-category, complete Segal spaces, etc...) nobody bothered to develop such a theory. In addition doing so defeat the initial motivation you talk about :  Most examples of these interesting sSet-enriched category you are mentioning. And the cofibrant replacement construction is not a very explicit construction - much more complicated that the functor from sSet-enriched categories to quasi-categories or complete Segal Spaces.
As mentioned by Zhen Lin, in both case, this makes the construction of "functors categories" extremely annoying, and this is probably the single most important construction in basic higher category theory.
A: This is a long comment. Simon's answer is correct, but there are other ways around the difficulty of forming the right internal Hom.
If $C$ is a full simplicial subcategory of a simplicial model category such that all objects of $B$ are fibrant and which is stable under the factorization of maps into trivial cofibration/fibration, then, the formation of the simplicial category of simplicial functors $Fun(A,C)$ for each simplicial category $A$ is compatible with Dwyer-Kan equivalences. In particular, there is no need any more to to take cofibrant resolutions of $A$. In particular, on may consider a simplicial category $B$ and take for $C$ the full simplicial subcategory of simplicial functors $B^{op}\to Kan$ with values in Kan complexes which are weakly equivalent to those of the form $Hom_B(-,b)$ for some object $b$ of $B$. In other words, we have a perfectly fine access to the $(\infty,1)$-category-theoretic Yoneda embedding in the context of simplicial categories, and it fits very well with its more rigid sSet-enriched counterpart. One can also express the theory of Kan extensions very naturally in the setting of sSet-enriched categories. In other words, if we think of objects of an $(\infty,1)$-category as representable functors, given two $(\infty,1)$-categories $A$ and $B$, we can associate their presheaf categories $\widehat A$ and $\widehat B$, respectively, and think of functors $A\to B$ as colimit preserving functors $\widehat A\to\widehat B$ that preserve representable presheaves. This point of view is very robust and has a nice interpretation in all models of $(\infty,1)$-categories, including sSet-categories. I would like to insist on the fact that this is the way the theory has been developed by Dwyer and Kan (all the assertions I made above without proof are in their papers). I would not qualify this approach as "annoying" - it is quite efficient, actually.  In fact this point of view is very robust and also applies to categories enriched in a (nice enough) model category; for instance, in the case of chain complexes, this leads to Bertrand Toen's paper The homotopy theory of dg-categories and derived Morita theory which gives a very workable approach (197 citations on Mathscinet).
Having access to the Yoneda embedding is indeed the central feature of $(\infty,1)$-category theory: this encodes straightening/unstraightening together with the theory of (pointwise) Kan extensions, out of which on can do everything else (abstract constructions as well as computations). Indeed, given a functor $p:X\to A$, we compose $p$ with the Yoneda embedding and take the colimit: this is "straightening". "Unstraightening" is the claim that "straightening" is a left Boudfield localization whose right adjoint takes values in cartesian fibrations whose fibers are $\infty$-groupoids. We could summarize that, in other models of $(\infty,1)$-categories (such as quasi-categories or complete Segal spaces), there is a nice theory (which looks like a literal interpretation the language of ordinary category theory) where the main difficulty is the construction of the universe (the $\infty$-category of $\infty$-groupoids), whereas in the model of simplicial categories, there is a rather easy access to the universe (Kan complexes naturally form a simplicial category) and to pointwise Kan extensions, but everything else is mode difficult. Lurie's take in Higher Topos Theory is to take advantage of both approaches to begin with.
A final comment about the approach of weak functors: if we weaken the notion of functor, we should weaken everyting, including the composition law. A way to do it is to weaken the notion of finite sequence of composable functors of length $n$, such as $A_0\to A_1\to\ldots\to A_n$, for each $n\geq 0$. The fun part is that we do not obtain a category, but a quasi-category of simplicial categories (we could do that by replacing categories with algebras over any coloured operad). This is how Boardmann and Vogt discovered the very notion of quasi-category (that they coined as weak Kan complexes). In other words, the first example of quasi-category that is not the nerve of a $1$-category comes from this idea of weakening!
