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 $\phi$.