Let $\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}$ be a family of pointed sets.
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Functoriality. The assignment $\webleft\{ \webleft (X_{i},x^{i}_{0}\webright )\webright\} _{i\in I}\mapsto \webleft (\prod _{i\in I}X_{i},\webleft (x^{i}_{0}\webright )_{i\in I}\webright )$ defines a functor
\[ \prod _{i\in I}\colon \mathsf{Fun}\webleft (I_{\mathsf{disc}},\mathsf{Sets}_{*}\webright )\to \mathsf{Sets}_{*}. \]
