> [!definition] > > Let $\mathfrak{A}$ be a [[Category|category]] and $A, B \in \mathfrak{A}$. *A* **product** of $A$ and $B$ is a triple $(P, \pi_A, \pi_B)$ where $P \in \mathfrak{A}$ and $\pi_A \in \text{Mor}(P, A)$, $\pi_B \in \text{Mor}(P, B)$ > > ```tikz > \usepackage{tikz-cd} > \begin{document} > \begin{tikzcd} & P \\ X && Y \arrow["{\pi_X}"', from=1-2, to=2-1] \arrow["{\pi_Y}", from=1-2, to=2-3] \end{tikzcd} > \end{document} > ``` > > such that given any morphisms $f_A: C \to A$ and $f_B: C \to B$, there exists a unique morphism $f: C \to P$ such that > > ```tikz > \usepackage{tikz-cd} > \begin{document} > \begin{tikzcd} & C \\ & P \\ X && Y \arrow["f", from=1-2, to=2-2] \arrow["{f_X}"', from=1-2, to=3-1] \arrow["{f_Y}", from=1-2, to=3-3] \arrow["{\pi_X}", from=2-2, to=3-1] \arrow["{\pi_Y}"', from=2-2, to=3-3] \end{tikzcd} > \end{document} > ``` > > More generally, let $\seqi{A}$ be a family of objects in $\mathfrak{A}$. A **product** of this family is the pair $(P, \seqi{\pi})$, where $P$ is an object in $\mathfrak{A}$ and $\pi_i: P \to A_i$, such that given a family of morphisms $\seqi{g}$ where $g_i: C \to A_i$, there exists a unique morphism $h: C \to P$ such that $f_i \circ h = g_i$ for all $i \in I$.