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## Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

Motivation

In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in $Cat$. One axiom of basic localizers asserts that every category which has a terminal object is "aspherical", i.e. the canonical arrow from it to the point is in any basic localizer. It is a sufficient condition to get all the properties we want, provided the other axioms hold.

I have recently worked on "2-basic localizers", the analogous classes of (let say strict for convenience) 2-functors. It appears that what seems, from this viewpoint, the right analogous notion of "category with terminal object" is "2-category which has an object $z$ such that, given any object $x$, the category $Hom (x,z)$ has a terminal object". Instances of such 2-categories are "slices over an object". (There are, of course, three dual notions, the four of them corresponding to the two ways to slice over an object and the two ways to slice under an object.)

Question 1

Does this property have a standard name?

I am aware that there should be something like an adjunction between such a 2-category and the terminal one going on here, but I am really looking for standard terminology if there is one now.

Question 2

Do this kind of 2-categories or this kind of property appear naturally in other contexts?

This second question may be as silly as asking "where do categories with a terminal object crop up", but category theorists I have talked to do not seem to have encountered such a notion. I hope it could ring a bell for other people, especially those working on homotopy-related stuff.

EDIT: To be a bit more precise, I have the feeling that this property has something to do with prefibrations in $2-Cat$. Perhaps I will tell more about that later.

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I don't know whether this is useful in your context, but an alternate definition of "category with a terminal object" you might consider is an object $z$ such that there exists an adjunction between $z$ and the terminal object of your 2-category. – Eric Wofsey Sep 29 at 18:19
Extending Eric's comment --- one may say, that an object $z$ has an internal terminal point if the unique arrow $z \rightarrow 1$ has a right adjoint $\top \colon 1 \rightarrow z$. By 2-Yoneda this means that the transformation $\hom(-, z) \rightarrow \hom(-, 1)$ has a right adjoint, which, modulo a kind of Beck-Chevalley condition (the stability condition), is equivalent to saying that for each $x$ the category $\hom(x, z)$ has a terminal object. – Michal R. Przybylek Sep 29 at 18:59
I appreciate that some people try to make others' questions better by fixing typos and LaTeX, correcting language mistakes, suggesting changes and so on, but, in this case, I would have liked to know why Andrej Bauer thought the title of my question would be better the way he has put it. In the past, I have edited the title of a question of mine after someone suggested I do so in a comment. If such a choice have been made here (suggesting instead of modifying arbitrarily), I could have replied (before the title was changed) that, for one thing, (continued) – Jonathan Chiche Sep 29 at 23:29
(continued) deleting question marks has the effect that the title is not a question anymore, and I want the title of my questions to be real questions. (The button on the right of my screen is called "Ask Question".) In addition, I want them to be precise. One of the reasons for this choice is that I would not like people to waste their time reading my questions if they could know by the title that they are not interested by them. The modification makes the title lose its precision as well as its character of being a question. Therefore, I am going back to the original one. (Continued.) – Jonathan Chiche Sep 29 at 23:40
Here is something for people to upvote or not: your title is too long, a shorter one would be better. – Andrej Bauer Sep 30 at 3:34
As a matter of general principle, homsets inherit properties of the target. This is not invariable, but just a guideline. It seems clear that in Cat, for every $z$ that has a terminal object, the category Hom$(x,z)$ will also have a terminal object since the functor that is constant at the terminal object will be terminal. I don't know if this is general (it is too late in the day) but that is where I would look.