ScalingStacks

There is a $2$-functor from $\mathrm{Alg}$ to $\mathrm{Bimod}$: it sends $A$ to the differential category ${\mathcal{C}}_{A}$ with one object $c_{A}$ and $\operatorname{End}\nolimits(c_{A})=A$. It sends an $(A^{\prime},A)$-bimodule $M$ to the $({\mathcal{C}}_{A^{\prime}},{\mathcal{C}}_{A})$-bimodule ${\mathcal{C}}_{M}$ given by ${\mathcal{C}}_{M}(c_{A},c_{A^{\prime}})=M$. This $2$-functor provides isomorphisms of categories $\operatorname{Hom}\nolimits_{\mathrm{Alg}}(A,A^{\prime})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{\mathrm{Bimod}}({\mathcal{C}}_{A},{\mathcal{C}}_{A^{\prime}})$.

There is a $2$-fully faithful $2$-functor from the $2$-category of differential categories to $\mathrm{Bimod}^{\mathrm{rev}}$: it sends ${\mathcal{C}}$ to ${\mathcal{C}}$ and $F:{\mathcal{C}}\to{\mathcal{C}}^{\prime}$ to the $({\mathcal{C}},{\mathcal{C}}^{\prime})$-bimodule $(c,c^{\prime})\mapsto\operatorname{Hom}\nolimits(c^{\prime},F(c))$.

There is a $2$-fully faithful $2$-functor from $\mathrm{Bimod}$ to the $2$-category of differential categories: it sends ${\mathcal{C}}$ to ${\mathcal{C}}\operatorname{\!-diff}\nolimits$ and $M$ a $({\mathcal{C}}^{\prime},{\mathcal{C}})$-bimodule to $M\otimes_{{\mathcal{C}}}-:{\mathcal{C}}\operatorname{\!-diff}\nolimits\to{\mathcal{C}}^{\prime}\operatorname{\!-diff}\nolimits$.

Composing the $2$-functor $\mathrm{Alg}\to\mathrm{Bimod}$ and the $2$-functor from $\mathrm{Bimod}$ to the $2$-category of differential categories, we obtain a differential $2$-functor from $\mathrm{Alg}$ to the $2$-category of differential categories: it sends $A$ to $A\operatorname{\!-diff}\nolimits$ and it sends an $(A^{\prime},A)$-bimodule $M$ to the functor $M\otimes_{A}-:A\operatorname{\!-diff}\nolimits\to A^{\prime}\operatorname{\!-diff}\nolimits$. Note that this $2$-functor is $2$-fully faithful.