Kerodon

Warning 6.2.2.2. Definition 6.2.2.1 is not self-dual. We say that a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is a $\operatorname{\mathcal{C}}'$-colocal equivalence if, for every object $W \in \operatorname{\mathcal{C}}'$, composition with $u$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \xrightarrow { [u] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)$. In other words, $u$ is a $\operatorname{\mathcal{C}}'$-colocal equivalence if it is a $\operatorname{\mathcal{C}}'^{\operatorname{op}}$-local equivalence when viewed as a morphism in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Beware that this is usually not equivalent to the requirement that $u$ is a $\operatorname{\mathcal{C}}'$-local equivalence (see Example 6.2.2.3).