Given an indexed collection of functions $\{f_i : X_i \to Y_i\}_{i\in I}$, there is a product map $\prod_{i\in I}f_i:\prod_{i\in I}X_i \to \prod_{i\in I}Y_i$ defined by $(x_i)_{i\in I} \mapsto (f_i(x_i))_{i\in I}$. This is, I believe, the standard notion of a product map, which makes the Cartesian product into a multifunctor on $\textbf{Set}$. However, there is a related notion, where if we have an indexed collection of functions $\{f_i : X \to Y_i\}_{i\in I}$ for some fixed domain $X$, there is a "product map" $\prod^*_{i\in I}f_i:X \to \prod_{i\in I}Y_i$ defined by $x \mapsto (f_i(x))_{i\in I}$. I am not sure what the standard name or standard notation for this second notion is. Can someone tell me what it is?
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Each of these can be viewed as a special case of the other. It seems the terminology is not totally standardised, which sometimes happens with category theory - see the answers here. – Izaak van Dongen Mar 10 '24 at 00:58
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It is the map you get due to the universal property of the product. A common notation for it is $(f_i)_{i \in I}$.
A less common name and notation appears in Engelking's General Topology: He calls it the diagonal of the mappings $\{f_i\}_{i \in I}$ and he denotes it by $\triangle_{i \in I} f_i$.
azif00
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