Universal properties of products and coproducts

Exercise 1.7.8. Let \(I\) be a set, and let \(\{x_i\}\) be a collection of objects of an \(\infty \)-category \(C\).

(1)

Show that an object \(x\) in \(C\) is a product of the objects \(x_i\) if and only if it comes equipped with morphisms \(\pr _i\colon x \to x_i\) such that for any other object \(y\) of \(C\) the map \[ (\pr _i \circ -)_i \colon \Hom _C(y,x) \to \prod _{i \in I} \Hom _C(y,x_i) \] is an equivalence of animae.

(2)

Formulate and prove the dual statement for coproducts.

Hint: show that there is an equivalence \(\Fun (I,C) \simeq \prod _{i \in I} C\).