Remark 9.5.4.11. In the situation of Variant 9.5.4.10, suppose that $\operatorname{\mathcal{D}}$ is a complete $\infty $-category. Then a functor $T: \operatorname{\mathcal{D}}\rightarrow \varprojlim (\mathscr {F})$ is continuous if and only if, for each $C \in \operatorname{\mathcal{C}}$, the composite functor
\[ T_ C: \operatorname{\mathcal{D}}\rightarrow \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C) \]
is continuous. Similarly, if $\operatorname{\mathcal{D}}$ is accessible, then $T$ is accessible if and only if each of the functors $T_{C}$ is accessible. See Proposition 7.6.6.24 and Remark 9.4.8.10.