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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.

HTT

Am I wrong? Where did I make a mistake?

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  • $\begingroup$ It seems to me that this map should exist as soon as $F$ is a functor, it doesn't need to be $S$-enriched. Am I missing something? $\endgroup$ – Arnaud D. Feb 6 '18 at 9:23
  • $\begingroup$ @ArnaudD. My thought: $F$ as a functor only defines $\operatorname{Hom}(X, GY) \to \operatorname{Hom}(FX, FGY)$, which is a morphism in $\mathsf{Set}$, not in $\mathbf{S}$. $\endgroup$ – B. W. Feb 6 '18 at 9:35
  • $\begingroup$ Ah, that's what I missed. Good point. $\endgroup$ – Arnaud D. Feb 6 '18 at 9:43
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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.

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