Construction 9.3.4.32 (Inverse Images). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits fiber products. Then every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ determines a pullback functor
(see Proposition 7.6.2.24). The functor $f^{\ast }$ has a left adjoint, and therefore carries subterminal objects of $\operatorname{\mathcal{C}}_{/X}$ to subterminal objects of $\operatorname{\mathcal{C}}_{/Y}$. Passing to isomorphism classes, we obtain a map of partially ordered sets $f^{-1}: \operatorname{Sub}(Y) \rightarrow \operatorname{Sub}(X)$, given concretely by the formula $f^{-1} [Y_0] = [ Y_0 \times _{Y} X ]$. Since the functor $f^{\ast }$ preserves products, $f^{-1}$ is a homomorphism of lower semilattices: that is, it satisfies the identities