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If $C$ is a category, its core is the subgroupoid consisting of the isomorphisms of $C$. This generalizes; if $\mathcal{C}$ is an $\infty$-category, we define its core is the $\infty$-subgroupoid consisting only of equivalences.

If $\mathcal{C}$ is a quasi-category, define $\mathrm{Core}(\mathcal{C})$ by the following pullback of simplicial sets:

$$ \require{AMScd} \begin{CD} \mathrm{Core}(\mathcal{C}) @>>> \mathcal{C} \\ @VVV @VVV \\ \mathbf{N}(\mathrm{Core}(\mathrm{h}\mathcal{C})) @>>> \mathbf{N}(\mathrm{h}\mathcal{C}) \end{CD} $$ The bottom horizontal map is an inner fibration because it is a functor between (nerves of) categories (introduction to HTT section 2.3). I think you can show the right vertical map is an inner fibration as well. Therefore, $\mathrm{Core}(\mathcal{C})$ is a quasi-category. (and furthermore, this diagram computes a pullback in the $\infty$-category of $\infty$-categories)

HTT propositions 1.2.5.1 and 1.2.5.3 together imply that $\mathrm{Core}(\mathcal{C})$ is the maximal Kan subcomplex of $\mathcal{C}$. And by construction, we can see a morphism of $\mathcal{C}$ is in $\mathrm{Core}(\mathcal{C})$ if and only if it is an isomorphism in the homotopy category.

(the remarks following the proof of 1.2.5.3 state this construction is actually right adjoint to the inclusion $\mathbf{Kan} \to \mathbf{QuasiCat}$)

Since $\Delta^1 \to I$ is a Kan equivalence, it follows that there is an equivalence of $\infty$-groupoids $$ \mathrm{Core}(\mathcal{C})^I \to \mathrm{Core}(\mathcal{C})^{\Delta^1} $$ and $\mathrm{Core}(\mathcal{C})^{\Delta^1} \subseteq \mathcal{C}^I$ is precisely the subcategory spanned by the functors mapping the arrow of $\Delta^1$ to an isomorphism of $\mathrm{h}\mathcal{C}$.

If $C$ is a category, its core is the subgroupoid consisting of the isomorphisms of $C$. This generalizes; if $\mathcal{C}$ is an $\infty$-category, we define its core is the $\infty$-subgroupoid consisting only of equivalences.

If $\mathcal{C}$ is a quasi-category, define $\mathrm{Core}(\mathcal{C})$ by the following pullback of simplicial sets:

$$ \require{AMScd} \begin{CD} \mathrm{Core}(\mathcal{C}) @>>> \mathcal{C} \\ @VVV @VVV \\ \mathbf{N}(\mathrm{Core}(\mathrm{h}\mathcal{C})) @>>> \mathbf{N}(\mathrm{h}\mathcal{C}) \end{CD} $$ The bottom horizontal map is an inner fibration because it is a functor between (nerves of) categories (introduction to HTT section 2.3). I think you can show the right vertical map is an inner fibration as well. Therefore, $\mathrm{Core}(\mathcal{C})$ is a quasi-category. (and furthermore, this diagram computes a pullback in the $\infty$-category of $\infty$-categories)

HTT propositions 1.2.5.1 and 1.2.5.3 together imply that $\mathrm{Core}(\mathcal{C})$ is the maximal Kan subcomplex of $\mathcal{C}$. And by construction, we can see a morphism of $\mathcal{C}$ is in $\mathrm{Core}(\mathcal{C})$ if and only if it is an isomorphism in the homotopy category.

(the remarks following the proof of 1.2.5.3 state this construction is actually right adjoint to the inclusion $\mathbf{Kan} \to \mathbf{QuasiCat}$)

Since $\Delta^1 \to I$ is a Kan equivalence, it follows that there is an equivalence of $\infty$-groupoids $$ \mathrm{Core}(\mathcal{C})^I \to \mathrm{Core}(\mathcal{C})^{\Delta^1} $$ and the objects of $\mathrm{Core}(\mathcal{C})^{\Delta^1} \subseteq \mathcal{C}^{\Delta^1}$ are precisely functors mapping the arrow of $\Delta^1$ to an isomorphism of $\mathrm{h}\mathcal{C}$.

If $C$ is a category, its core is the subgroupoid consisting of the isomorphisms of $C$. This generalizes; if $\mathcal{C}$ is an $\infty$-category, we define its core is the $\infty$-subgroupoid consisting only of equivalences.

If $\mathcal{C}$ is a quasi-category, define $\mathrm{Core}(\mathcal{C})$ by the following pullback of simplicial sets:

$$ \require{AMScd} \begin{CD} \mathrm{Core}(\mathcal{C}) @>>> \mathcal{C} \\ @VVV @VVV \\ \mathbf{N}(\mathrm{Core}(\mathrm{h}\mathcal{C})) @>>> \mathbf{N}(\mathrm{h}\mathcal{C}) \end{CD} $$ The bottom horizontal map is an inner fibration because it is a functor between (nerves of) categories (introduction to HTT section 2.3). I think you can show the right vertical map is an inner fibration as well. Therefore, $\mathrm{Core}(\mathcal{C})$ is a quasi-category. (and furthermore, this diagram computes a pullback in the $\infty$-category of $\infty$-categories)

HTT propositions 1.2.5.1 and 1.2.5.3 together imply that $\mathrm{Core}(\mathcal{C})$ is the maximal Kan subcomplex of $\mathcal{C}$. And by construction, we can see a morphism of $\mathcal{C}$ is in $\mathrm{Core}(\mathcal{C})$ if and only if it is an isomorphism in the homotopy category.

(the remarks following the proof of 1.2.5.3 state this construction is actually right adjoint to the inclusion $\mathbf{Kan} \to \mathbf{QuasiCat}$)

Your question can be reinterpreted as asking whether both maps in the following span are weak equivalences when $\mathcal{C}$ is a quasicategory: $$ \mathcal{C}^I \leftarrow \mathrm{Core}(\mathcal{C})^I \to \mathrm{Core}(\mathcal{C})^{\Delta^1} $$

Since $\Delta^1 \to I$ is a Kan equivalence, it follows that the rightward map is an equivalence of $\infty$-groupoids.

If $C$ is a category, its core is the subgroupoid consisting of the isomorphisms of $C$. This generalizes; if $\mathcal{C}$ is an $\infty$-category, we define its core is the $\infty$-subgroupoid consisting only of equivalences.

If $\mathcal{C}$ is a quasi-category, define $\mathrm{Core}(\mathcal{C})$ by the following pullback of simplicial sets:

$$ \require{AMScd} \begin{CD} \mathrm{Core}(\mathcal{C}) @>>> \mathcal{C} \\ @VVV @VVV \\ \mathbf{N}(\mathrm{Core}(\mathrm{h}\mathcal{C})) @>>> \mathbf{N}(\mathrm{h}\mathcal{C}) \end{CD} $$ The bottom horizontal map is an inner fibration because it is a functor between (nerves of) categories (introduction to HTT section 2.3). I think you can show the right vertical map is an inner fibration as well. Therefore, $\mathrm{Core}(\mathcal{C})$ is a quasi-category. (and furthermore, this diagram computes a pullback in the $\infty$-category of $\infty$-categories)

HTT propositions 1.2.5.1 and 1.2.5.3 together imply that $\mathrm{Core}(\mathcal{C})$ is the maximal Kan subcomplex of $\mathcal{C}$. And by construction, we can see a morphism of $\mathcal{C}$ is in $\mathrm{Core}(\mathcal{C})$ if and only if it is an isomorphism in the homotopy category.

(the remarks following the proof of 1.2.5.3 state this construction is actually right adjoint to the inclusion $\mathbf{Kan} \to \mathbf{QuasiCat}$)

Since $\Delta^1 \to I$ is a Kan equivalence, it follows that there is an equivalence of $\infty$-groupoids $$ \mathrm{Core}(\mathcal{C})^I \to \mathrm{Core}(\mathcal{C})^{\Delta^1} $$ and $\mathrm{Core}(\mathcal{C})^{\Delta^1} \subseteq \mathcal{C}^I$ is precisely the subcategory spanned by the functors mapping the arrow of $\Delta^1$ to an isomorphism of $\mathrm{h}\mathcal{C}$.

If $C$ is a category, its core is the subgroupoid consisting of the isomorphisms of $C$. This generalizes; if $\mathcal{C}$ is an $\infty$-category, we define its core is the $\infty$-subgroupoid consisting only of equivalences.

If $\mathcal{C}$ is a quasi-category, define $\mathrm{Core}(\mathcal{C})$ by the following pullback of simplicial sets:

$$ \require{AMScd} \begin{CD} \mathrm{Core}(\mathcal{C}) @>>> \mathcal{C} \\ @VVV @VVV \\ \mathbf{N}(\mathrm{Core}(\mathrm{h}\mathcal{C})) @>>> \mathbf{N}(\mathrm{h}\mathcal{C}) \end{CD} $$ The bottom horizontal map is an inner fibration because it is a functor between (nerves of) categories (introduction to HTT section 2.3). I think you can show the right vertical map is an inner fibration as well. Therefore, $\mathrm{Core}(\mathcal{C})$ is a quasi-category. (and furthermore, this diagram computes a pullback in the $\infty$-category of $\infty$-categories)

HTT propositions 1.2.5.1 and 1.2.5.3 together imply that $\mathrm{Core}(\mathcal{C})$ is the maximal Kan subcomplex of $\mathcal{C}$. And by construction, we can see a morphism of $\mathcal{C}$ is in $\mathrm{Core}(\mathcal{C})$ if and only if it is an isomorphism in the homotopy category.

(the remarks following the proof of 1.2.5.3 state this construction is actually right adjoint to the inclusion $\mathbf{Kan} \to \mathbf{QuasiCat}$)

Your question can be reinterpreted as asking whether both maps in the following span are weak equivalences when $\mathcal{C}$ is a quasicategory: $$ \mathcal{C}^I \leftarrow \mathrm{Core}(\mathcal{C})^I \to \mathrm{Core}(\mathcal{C})^{\Delta^1} $$

Since $\Delta^1 \to I$ is a Kan equivalence, it follows that the rightward map is an equivalence of $\infty$-groupoids.

I'm pretty sure the leftward map is actually an isomorphism of simpilcial sets, but I don't see off hand an easy way to internalize the fact that $\hom(K, \mathrm{Core}(\mathcal{C})) \to \hom(K, \mathcal{C})$ is a natural bijection when restricted to Kan complexes (given by 1.2.5.3).

If $C$ is a category, its core is the subgroupoid consisting of the isomorphisms of $C$. This generalizes; if $\mathcal{C}$ is an $\infty$-category, we define its core is the $\infty$-subgroupoid consisting only of equivalences.

If $\mathcal{C}$ is a quasi-category, define $\mathrm{Core}(\mathcal{C})$ by the following pullback of simplicial sets:

$$ \require{AMScd} \begin{CD} \mathrm{Core}(\mathcal{C}) @>>> \mathcal{C} \\ @VVV @VVV \\ \mathbf{N}(\mathrm{Core}(\mathrm{h}\mathcal{C})) @>>> \mathbf{N}(\mathrm{h}\mathcal{C}) \end{CD} $$ The bottom horizontal map is an inner fibration because it is a functor between (nerves of) categories (introduction to HTT section 2.3). I think you can show the right vertical map is an inner fibration as well. Therefore, $\mathrm{Core}(\mathcal{C})$ is a quasi-category. (and furthermore, this diagram computes a pullback in the $\infty$-category of $\infty$-categories)

HTT propositions 1.2.5.1 and 1.2.5.3 together imply that $\mathrm{Core}(\mathcal{C})$ is the maximal Kan subcomplex of $\mathcal{C}$. And by construction, we can see a morphism of $\mathcal{C}$ is in $\mathrm{Core}(\mathcal{C})$ if and only if it is an isomorphism in the homotopy category.

(the remarks following the proof of 1.2.5.3 state this construction is actually right adjoint to the inclusion $\mathbf{Kan} \to \mathbf{QuasiCat}$)

Your question can be reinterpreted as asking whether both maps in the following span are weak equivalences when $\mathcal{C}$ is a quasicategory: $$ \mathcal{C}^I \leftarrow \mathrm{Core}(\mathcal{C})^I \to \mathrm{Core}(\mathcal{C})^{\Delta^1} $$

Since $\Delta^1 \to I$ is a Kan equivalence, it follows that the rightward map is an equivalence of $\infty$-groupoids.