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In this paper, we prove the standard comparison used by mathematicians between the idempotent complete pretriangulated dg-categories, over a unitary and commutative ring $k$, and the idempotent complete $k$-linear stable $\infty$-categories. Our approach is completely included in the $\infty$-categorical theory. To achieve the target we will reinterpret the Morita theory for dg-categories and we set the Morita theory for $k$-linear stable $\infty$-category.

We establish the feasibility of investigating the theory of $R\text{-}\mathrm{Mod}$-enriched categories, for any commutative and unitary ring $R$, through the framework of $\mathbb{A}\mathrm{b}$-enriched category theory. In particular, we prove that the category of $R$-$\mathrm{Mod}$-enriched categories, $Cat(R$-$\mathrm{Mod})$, the category of $\underline{R}$-modules inside $Cat(\mathbb{A}\mathrm{b})$, $\mathrm{LMod}_{\underline{R}}(Cat(\mathbb{A}\mathrm{b}))$, and the category of $Cat(\mathbb{A}\mathrm{b})$-enriched functors, $Fun^{Cat(\mathbb{A}\mathrm{b})}(\underline{\underline{R}},Cat(\mathbb{A}\mathrm{b}))$ are equivalent.

We establish the feasibility of investigating the theory of $\mathrm{Mod}_{\mathbb{H}\mathrm{k}}$-enriched $\infty$-categories, where $\mathbb{H}\mathrm{k}$ is the Eilenberg-Maclane Spectrum associated with a commutative and unitary ring $k$, through the framework of $\mathcal{S}p$-enriched $\infty$-category theory. In particular, we prove that the $\infty$-category of $\mathrm{Mod}_{\mathbb{H}\mathrm{k}}$-enriched $\infty$-categories $\mathcal{C}at_{\infty}^{\mathrm{Mod}_{\mathbb{H}\mathrm{k}}}$, $\infty$-category of left $\mathbb{H}\mathrm{k}$-module objects of the $\infty$-category of $\mathcal{S}p$-enriched $\infty$-categories $\mathcal{C}at_{\infty}^{\mathcal{S}p}$ $\mathrm{LMod}_{\mathbb{H}\mathrm{k}}(\mathcal{C}at_{\infty}^{\mathcal{S}p})$ and the $\infty$-category of $\mathcal{C}at_{\infty}^{\mathcal{S}p}$-enriched $\infty$-functors $Fun^{\mathcal{C}at_{\infty}^{\mathcal{S}p}}(\underline{\underline{\mathbb{H}\mathrm{k}}},\mathcal{C}at_{\infty}^{\mathcal{S}p})$ are equivalent.