CatGloss

Let $\mathscr{A}$ be a category, $I$ a set, and $(X_i){i \in I}$a family of objects of$\mathscr{A}$. A product of $(X_i){i \in I}$consists of an object$P$ and a family of maps \(\Bigl(P \stackrel{p_i}{\longrightarrow} X_i\Bigr)_{i \in I}\) with the property that for all objects $A$ and families of maps \(\Bigl(A \stackrel{f_i}{\longrightarrow} X_i\Bigr)_{i \in I}\) there exists a unique map $\bar{f}{\colon}\linebreak[0] A \to P$such that$p_i \circ \bar{f} = f_i$for all$i \in I$.

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