Product

Remarks

A product is the limit of the diagram whose shape is the set $I$ seen as a category with only identity morphisms. So it is unique up to unique isomorphism, and the product of the empty family is a terminal object.

Example: in the category of sets, the Cartesian product $X_1 \times X_2$ with the coordinate maps is a product. Given $f_1 \colon Y \to X_1$ and $f_2 \colon Y \to X_2$, the map $f(y) = (f_1(y), f_2(y))$ satisfies $\pi_i \circ f = f_i$, and any such $f$ must have these coordinates. In the category of groups the product is the direct product, and in the category of topological spaces it is the Cartesian product with the product topology.

Non-example: in the category of fields, $\mathbb{F}_2$ and $\mathbb{F}_3$ have no product. Field homomorphisms preserve the characteristic, so no field $P$ has morphisms to both $\mathbb{F}_2$ and $\mathbb{F}_3$.