The most basic formulation of a universal property

Question

I'm reading Riehl's category book, and she says in the first section of the second chapter (I quote): The most basic formulation of a universal property is to say that a particular object defines an initial or terminal object in its ambient category.

In what sense is an initial or terminal object universal in the category it lives in? is it perhaps regarding the idea that, for instance, an initial object has a unique form of perceiving every object in the ambient category?...

Edit: The universal property I know is about universal morphisms from an object to a functor, which can be seen as initial or terminal objects in a comma category, depending on the ambient category. But here, we are not considering a functor but only a category. Because of it I'm very confused about the notion of universality here.

Answer 1

1. It is universal in the sense that it has a universal property: given any object $X$, there is a unique morphism into $X$. And given any object with morphisms into every object, there is a an isomorphism between them. In fact, the initial object is a limit: the limit of the identity functor, and so has the universal property of the limit; and it is also a colimit: the colimit of the empty functor; so it has the universal property of the colimit and is a universal object relative to that colimit.

2. Likewise, the terminal object is universal: there are (unique) morphism from every object into it; and given any object that has morphisms from every object to it, there is a (unique) isomorphism between them. The terminal object is the colimit of the identity functor; and it is also the limit of the empty functor.

Universal objects can usually be described as initial or terminal objects in auxiliary categories. For example, the product of the family $\{X_i\}_{i\in I}$ in the category $\mathcal{C}$ is a terminal object in the category whose elements are ordered pairs $(Y,\{f_i\}_{i\in I})$, where $Y$ is an object of $\mathcal{C}$, $f_i\in\mathcal{C}(Y,X_i)$ for each $i$; and morphisms $f\colon (Y,\{f_i\})\to (Z,\{g_i\})$ are elements of $\mathcal{C}(Y,Z)$ such that for each $i$, $f_i=g_i\circ f$.