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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.

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  • $\begingroup$ It's not so much the (up to isomorphism) initial object, but rather the property of "initialness", which is universal. In a given category the initial object (if it exists, and again up to isomorphism) is the unique (up to isomorphism) object satisfying the property of "initialness" within that category. (And same with terminal objects vs. "terminality.") $\endgroup$ – Noah Schweber Apr 10 at 1:00
  • $\begingroup$ @NoahSchweber It is precisely my doubt, Why is the property of initialness called universal? $\endgroup$ – John Mars Apr 10 at 1:03
  • $\begingroup$ @NoahSchweber 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. $\endgroup$ – John Mars Apr 10 at 1:12
  • $\begingroup$ Your most recent comment significantly clarifies your question for me; I think you should put it in the body of your post. $\endgroup$ – Noah Schweber Apr 10 at 1:15
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    $\begingroup$ An initial object is the colimit of the empty functor, with the maps from the initial object to other objects being the “unique morphism” that exists because of the universal property of the colimit. It is also the limit of the identity functor, in which case the “unique morphisms” are the projection maps associated to the cone given by the limit and the functor. You have two functors to choose from to associate to the initial object, and similarly for the terminal one. $\endgroup$ – Arturo Magidin Apr 10 at 1:38
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  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 $i$, $f_i=g_i\circ f$.

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  • $\begingroup$ Relating to $1.$, why is this property called universal? $\endgroup$ – John Mars Apr 10 at 1:05
  • $\begingroup$ @JohnMars: I don’t understand the question. It is exactly the kind of property that is called universal. It has a property and a unique morphism into any other object having the property: it has a cone from the empty functor to it, and a unique morphism respecting the cones into any object that has a cone from the empty functor to it. What do you think this isn’t a universal property just like the universal property of any colimit, product, coproduct, etc? $\endgroup$ – Arturo Magidin Apr 10 at 1:28

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