Definition 7.6.2.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We will say that a functor $f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X}$ is given by pullback along $f$ if it is a right adjoint to the functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ given by postcomposition with $f$ (see Example 4.3.6.15). Note that this condition characterizes the functor $f^{\ast }$ up to isomorphism (see Remark 6.2.1.19).
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