Let $C$ and $D$ be presentable stable $\infty$-categories, and let $f:C \to D$ be a continuous functor between them. Let $0$ be the trivial stable $\infty$-category. What is the colimit of the diagram


$$0 \leftarrow C \rightarrow D$$


in the $\infty$-category of presentable stable $\infty$-categories and continuous functors? For example, is it a localization of $D$?



What is the cokernel of a map of presentable stable $\infty$-categories?
Yes. Colimits in $Pr^L$ are the same as limits in $Pr^R$, which are created by the forgetful functor to $Cat\_\infty$ (Higher Topos Theory, 5.5.3.18). So the pushout $E$ of your diagram is the pullback, in $Cat\_\infty$, of the diagram composed of the right adjoint functors. This means an object in $E$ is an object in $D$ together with an equivalence between its image in $C$ and the zero object, with the obvious mapping spaces. But the space of equivalences between an object and a final object, if not empty, is contractible. So the "extra data" of the equivalence is in fact no extra data at all, so that $E$ is a full subcategory of $D$ (the preimage of the zero object of $C$), and the map $D\to E$ is left adjoint to the inclusion.


