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Let $\mathcal A, \mathcal B$ be dg-categories over a field $k$. I denote by $\mathcal{RHom}(\mathcal A,\mathcal B)$ the dg-category (defined up to quasi-equivalence) which gives the internal hom in $\mathrm{Hqe}$ (see, for example, Keller's survey, Theorem 4.5). There is a natural $H^0$ functor: \begin{equation} H^0(-) :H^0(\mathcal{RHom}(\mathcal A,\mathcal B)) \to \mathcal{Hom}(H^0(\mathcal A),H^0(\mathcal B)), \end{equation} where $\mathcal{Hom}(-,-)$ here denotes the category of ordinary ($k$-linear) functors; $H^0(\mathcal{RHom}(\mathcal A,\mathcal B))$ can be identified to the category of quasi-functors $\mathrm{rep}(\mathcal A,\mathcal B)$.

I've been studying the properties of this functor for a while. For example, I know that for $\mathcal A = \mathbf{1}$, the category with one object and endomorphism ring $k$, $H^0(-)$ is an equivalence; for $\mathcal A = \Delta^1$, the category freely generated over $k$ by the diagram $0 \to 1$, this functor is full and reflects isomorphisms. Assuming $\mathcal B$ is pretriangulated and taking $\mathcal A = (\Delta^1)^{\mathrm{pretr}}$ (the pretriangulated hull of $\Delta^1$) the above claim implies that $H^0(-)$ induces, in this particular case, a bijection between the isomorphism classes of objects.

Do you know other situations where the $H^0$ functor exhibits some good properties as above? For example, are there other cases in which it is full and reflects isomorphisms? Or cases in which it induces a bijection between isomorphism classes of objects?

(Also, I was wondering if a similar problem is studied in the framework of $(\infty,1)$-categories.)

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In the framework of $\infty$-categories, I think it is not difficult to see that your claim for $\mathcal{A} = \Delta^1$ holds for all $\infty$-groupoids.

Let $K$ be an $\infty$-groupoid (Kan complex) and $C$ an $\infty$-category (weak Kan complex). The canonical map $C \to N h C$ is an isofibration (or fibration in the Joyal model structure), where $h$ is the homotopy category functor, left adjoint to the nerve functor $N$. This implies that 1) the induced map $$ \underline{\mathrm{Hom}}(K, C) \longrightarrow \underline{\mathrm{Hom}}(K, NhC) \times_{NhC} C $$ is a trivial Kan fibration, and 2) the canonical map $$ \underline{\mathrm{Hom}}(K, NhC) \times_{NhC} C \longrightarrow \underline{\mathrm{Hom}}(K, NhC) $$ induces a full, conservative, bijective-on-objects functor on the homotopy categories. Since $K$ is a Kan complex, one can prove that $$ N \underline{\mathrm{Hom}}_{\mathrm{Cat}}(hK, hC) = \underline{\mathrm{Hom}}(NhK, NhC) \longrightarrow \underline{\mathrm{Hom}}(K, NhC) $$ is a trivial Kan fibration. Now after passing to homotopy categories one gets a full and conservative functor $$ h\underline{\mathrm{Hom}}(K, C) \longrightarrow \underline{\mathrm{Hom}}_{\mathrm{Cat}}(hK, hC). $$

I am not sure whether the assumption on $K$ Is really necessary.

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