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In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the following:

Let $\mathbf{A}$ be a combinatorial simplicial model category, $S$ a small simplicial set, and $\phi:\mathfrak{C}[S]\to \mathcal{C}$ a DK-equivalence of small simplicial categories. Then the map $$\theta:N((\mathbf{A}^\mathcal{C})^\circ)\to \operatorname{Fun}(S,N(\mathbf{A}^\circ))$$is an equivlence of $\infty$-categories.

Here for a simplicial model category $\mathbf{B}$, we denoted by $\mathbf{B}^\circ $ its full simplicial subcategory spanned by the fibrant-cofibrant objects. Skimming through the proof, it looks like $\mathbf{A}^\mathcal{C}$ is endowed with the projective model structure. However, in Remark 4.2.4.5, he asserts that the claim also holds for the injective model structure. As far as I could tell, the proof does not easily generalize to the injective model structure. (If we replace "projective" by "injective" in Appendix A.3.4, the argument in the proof of Lemma A.3.4.10 does not hold.) Can someone explain to me why we can work with the injective model structure?

Any help/comment is appreciated. Thanks in advance.


Remark

The map $\theta$ is adjoint to the map $$N((\mathbf{A}^{\mathcal{C}})^{\circ})\times S\xrightarrow{1\times\psi}N((\mathbf{A}^{\mathcal{C}})^{\circ})\times N(\mathcal{C})\xrightarrow{N(\mathrm{ev})}N(\mathbf{A}^{\circ}),$$ where $\psi$ is the adjoint of $\phi$.

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    $\begingroup$ The first question is why there is such a map, in either model structure. I suspect that once this question is answered carefully, the question of why it is an equivalence will be seen to transfer from one model structure to the other. $\endgroup$
    – Tim Campion
    Feb 17, 2023 at 12:34
  • $\begingroup$ @TimCampion Yes, I should have explained how the comparison map is constructed. I added the necessary detail. $\endgroup$
    – Ken
    Feb 18, 2023 at 0:49

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As shown by Dwyer–Kan (Function complexes in homotopical algebra, Proposition 4.8), for a simplicial model category $A$, the simplicial category $A^\circ$ is Dwyer–Kan equivalent to the simplicial category given by the hammock localization of $A$ with respect to the weak equivalences of $A$.

Applying this fact to the model category $A^C$ equipped with its projective or injective model structure, and using the fact that weak equivalences are the same for both model structures, we arrive at the desired result.

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    $\begingroup$ Thanks! It does show that $N((\mathbf{A}^{\mathcal{C}})_{\mathrm{proj}}^{\circ})$ and $N((\mathbf{A}^{\mathcal{C}})_{\mathrm{inj}}^{\circ})$ can be joined by a zig-zag of weak categorical equivalences. However, I do not yet understand why the map $$ N((\mathbf{A}^{\mathcal{C}})_{\mathrm{inj}}^{\circ})\to\operatorname{Fun}(S,N(\mathbf{A}^{\circ})) $$ is an equivalence of $\infty$-categories. Can you elaborate on this? $\endgroup$
    – Ken
    Feb 16, 2023 at 10:11

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