I was reading the lecture notes of Peter Scholze on condensed mathematics and complex geometry. There is some notation there that is confusing to me - likely becuase I know very little about infinity/derived categories.
On page 48/49 Scholze is considering a symmetric monoidal stable $\infty$-category $\mathcal C$, and for an idempotent algebra object $A \in \mathcal C$ he defines the category $\mathcal C(U) = C/\text{Mod}_A(C)$ as the Verdier quotient. (Here $U$ represents the complement of the closed set defined by $A$ in the locale of idempotent algebra objects, and Mod$_A(C)$ is the subcategory of objects $X$ such that $X \cong X \otimes A$.)
The problem is that when Scholze writes that the localization functor $j_U^* : \mathcal C \to \mathcal C/ \text{Mod}_A(\mathcal C)$ comes with a left adjoint $j_{U!}$ and a right adjoint $j_{U*}$, which are determined by $j_{U!}j_U^* X = [X \to X \otimes A]$ and $j_{U*}j_U^* X = \underline{\text{RHom}}_C([1 \to A], X)$. I'm not quite sure what is meant by this. How is $[1 \to A]$ or $[X \to X \otimes A]$ meant to be an object of $\mathcal C$ at all?