Definiton of the beta in Lurie's HTT In the section §A.3.1 of Lurie's "Higher Topos Theory",
the map $ \beta_{X,S} \colon S \otimes FX \to F(S \otimes X) $
is defined without assuming $ F $ has
the structure of $ \mathbf{S} $-enriched functor.
However, in the definition, the morphism
$ \operatorname{Map}(X, GY) \to \operatorname{Map}(FX, FGY) $ seems to be used.

Am I wrong?
Where did I make a mistake?
 A: You are right that the quoted definition of the natural transformation $\beta$ uses an $\mathbf{S}$-enriched functor structure for $F$, in contradiction with the stated hypotheses. The source of the error is that the natural transformation $\beta$ should point in the opposite direction to what Lurie writes, i.e. it should have components $\beta_{S,X} \colon F(S\otimes X) \to S \otimes FX$. This direction for $\beta$ is in fact what is used in the proof of Proposition A.3.1.10 in HTT.
We may then correctly define $\beta_{S,X} \colon F(S\otimes X) \to S \otimes FX$ as the morphism corresponding under the Yoneda embedding to the composite natural transformation
\begin{align}
\mathcal{D}(S\otimes FX,Y) &\cong \mathbf{S}(S,\underline{\mathcal{D}}(FX,Y)) \\ 
&\to \mathbf{S}(S,\underline{\mathcal{C}}(GFX,GY)) \\ 
&\cong \mathcal{C}(S\otimes GFX,GY)  \\
&\to \mathcal{C}(S \otimes X,GY) \\ 
&\cong \mathcal{D}(F(S\otimes X),Y)
\end{align}
where the first arrow is given by the $\mathbf{S}$-enrichment of the functor $G$, the second arrow is given by composition with the unit $X \to GFX$ of the adjunction, and the isomorphisms are given by the universal properties of tensoring with the simplicial set $S$ and of the adjunction $F \dashv G$. (Note that $\mathcal{C}(-,-)$ and $\underline{\mathcal{C}}(-,-)$ denote the $\mathbf{Set}$-enriched and $\mathbf{S}$-enriched hom objects respectively.)
This definition can be understood in the context of categories bearing an action of the monoidal category $\mathbf{S}$, called "$\mathbf{S}$-actegories". The $\mathbf{S}$-enrichment of the functor $G$ gives it the structure of a "lax morphism" of $\mathbf{S}$-actegories, consisting of a natural transformation $S \otimes GY \to G(S \otimes Y)$ subject to axioms. By the theory of doctrinal adjunction, this lax morphism structure on $G$ corresponds to an "oplax morphism" structure on its left adjoint $F$, consisting of a natural transformation $F(S \otimes X) \to S \otimes FX$ subject to axioms, which is none other than the natural transformation $\beta$ defined above.
