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Example 8.4.4.5 (Functoriality of the Presheaf Construction). Let $\kappa $ be an uncountable regular cardinal, let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Assume that $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and that $\operatorname{\mathcal{D}}$ is locally $\kappa $-small, and fix covariant Yoneda embeddings

\[ h_{\bullet }^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \quad \quad h_{\bullet }^{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ). \]

Let $G: \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa } )$ be given by precomposition with $f$. Then the functor $G$ admits a left adjoint $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{<\kappa })$. Moreover, the diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{C}}} } \ar [d]^{f} & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \ar [d]^{F} \\ \operatorname{\mathcal{D}}\ar [r]^-{ h_{\bullet }^{\operatorname{\mathcal{D}}} } & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ). } \]

commutes up to isomorphism. This follows by applying Variant 8.4.4.4 to the composite functor $(h_{\bullet }^{\operatorname{\mathcal{D}}} \circ f): \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{D}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.