Kerodon

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Definition 7.6.1.1. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes and let $\operatorname{\mathcal{C}}$ be an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category. We say that a collection of morphisms $\{ q_ i: Y \rightarrow Y_ i \} _{i \in I}$ of $\operatorname{\mathcal{C}}$ exhibits $Y$ as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched product of the collection $\{ Y_ i \} _{i \in I}$ if, for every object $X \in \operatorname{\mathcal{C}}$, the collection of maps $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { q_ i \circ } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y_ i)$ induces an isomorphism

\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \prod _{i \in I} \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y_ i ) \]

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.

We say that a collection of morphisms $\{ e_ i: Y_ i \rightarrow Y \} _{i \in I}$ exhibits $Y$ as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched coproduct of the collection $\{ Y_ i \} _{i \in I}$ if, for every object $Z \in \operatorname{\mathcal{C}}$, the collection of maps $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ e_ i } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y_ i,Z)$ induces an isomorphism

\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \prod _{i \in I} \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y_ i ) \]