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Exercise 8.1.2.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $u: X' \rightarrow X$ and $v: Y \rightarrow Y'$, so that covariant transport for the left fibration $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ determines a morphism of Kan complexes

\[ T: \{ X\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow \{ X'\} \times _{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{Tw}(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y'\} . \]

Show that, under the identifications supplied by Corollary 8.1.2.7, the induced map of connected components $\pi _0(T): \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) \rightarrow \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X',Y')$ is given by the construction $[f] \mapsto [v] \circ [f] \circ [u]$.