Can you always pullback accessible t-structures along left/right adhoints between presentable $\infty$-categories?
Let $F: \mathcal{C} \to \mathcal{D}$ be a left adjoint functor of presentable $\infty$-categories (preserving $\kappa$-compact objects for some $\kappa$) and let $\mathcal{D}_{\ge 0}$ be the connective part of an accessible t-structure on $\mathcal{D}$. Then I want to say that $\mathcal{C}_{\ge 0} = \{X \in \mathcal{C} | F(X) \in \mathcal{D}_{\ge 0}\}$ is always the connective part of a t-structure on $\mathcal{C}$.
Let $F: \mathcal{C} \to \mathcal{D}$ be a left adjoint functor of presentable $\infty$-categories (preserving $\kappa$-compact objects for some $\kappa$) and let $\mathcal{D}_{\ge 0}$ be the connective part of an accessible t-structure on $\mathcal{D}$. Then I want to say that $\mathcal{C}_{\ge 0} = \{X \in \mathcal{C} | F(X) \in \mathcal{D}_{\ge 0}\}$ is always the connective part of a t-structure on $\mathcal{C}$.
