Universal characterization of the arrow (interval, ordinal, 1-simplex) object? (Is $2$-category theory actually needed?) In the "category of categories" $Cat$ (see e.g. here for a rigorous definition), there is a category $A$ with two objects and exactly one non-identity morphism. (Cf. "two-valued object", "bi-pointed object", and "cospan".)
Specifically, given the terminal category $\mathbf{1}$, there are exactly two distinct functors $\mathbf{1} \to A$, call them $s$ ("source") and $t$ ("target"), which of course correspond to the objects of $A$, and then the unique non-identity morphism within $A$ is denoted $s \overset{a}{\rightarrow} t$ ($a$ for "arrow").
Note that for any category $\mathcal{C}$, every morphism in $\mathcal{C}$ corresponds to a functor $A \to \mathcal{C}$. Moreover, building on the Cartesian closed structure of $Cat$, for any category $\mathcal{C}$ the exponential object/category $\mathcal{C}^A$ exists, and is usually called the "arrow category of $\mathcal{C}$ and denoted $Arr(\mathcal{C})$ or $\mathcal{C}^{\rightarrow}$ (cf. here, here, or here).
The category $A$ is sometimes denoted $\mathbf{2}$, although this risks confusion with the coproduct $\mathbf{1} \oplus \mathbf{1}$. Likewise, I have also seen it referred to as the "interval category", "the ordinal", or the "$1$-simplex", although all of those terms are ambiguous in the sense that they can (and more often do) refer to distinct notions than "the" category $A$.

Questions:

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*Does the category $A$ admit a characterization in terms of a universal property inside of $Cat$, allowing one to attempt to define analogous "arrow objects" in any category, e.g. $Set$ or $Top$? If so, is it a (co)limit, or another universal construction? Are there references describing this?


*Does the category $A$ (defined in terms of such a universal property) possess analogues in other categories? Again, are there references describing this?

Attempt:
1. I claim that the morphism $a$ can be identified with the endofunctor $\alpha: A \to A$ such that (abusing notation by identifying objects with functors from $\mathbf{1}$) $\alpha(s) = t$, $\alpha(t)=t$, and sending all morphisms to $id_t$.
The category $A$ can be written using the diagram (also if anyone can format this better please edit the post to do so)
$$\begin{matrix}  A & \overset{\alpha}{\longrightarrow} & A \\
s \nwarrow & & \nearrow t \\
& \mathbf{1} & \end{matrix} $$
I then claim that $(A, s, t, \alpha)$ is universal in the following sense, namely given any other category $\mathcal{C}$ with two objects $C_1$ and $C_2$ with an endofunctor $F: \mathcal{C} \to \mathcal{C}$ that sends $C_1$ to $C_2$, i.e. $F(C_1) = C_2$, as in the following diagram:
$$\begin{matrix}  \mathcal{C} & \overset{F}{\longrightarrow} & \mathcal{C} \\
C_1 \nwarrow & & \nearrow C_2 \\
& \mathbf{1} & \end{matrix} $$
there is a unique functor $I: A \to \mathcal{C}$ such that $I \circ \alpha = F \circ I$ and $I \circ s = C_1$, namely the following commutes:
$$\begin{matrix} A & & \overset{\alpha}{\longrightarrow} & & A \\
\vdots&s \nwarrow & & \nearrow t & \vdots \\
\exists! I &  & \mathbf{1} & & \exists! I \\
\downarrow & C_1 \swarrow & & \searrow C_2 &\downarrow \\
\mathcal{C} & & \overset{F}{\longrightarrow} & & \mathcal{C} \end{matrix}$$
If so, this would make $A$ some "initial-like" object, even if not exactly a colimit.
Possible problems with attempted solution for 1.
This doesn't seem quite right because it might be conflating endofunctors and morphisms too much (but we can't compose a morphism inside of a category with a functor between categories in a commutative diagram, right?) and also, if there are multiple morphisms $C_1 \to C_2$ inside of $\mathcal{C}$, then there should be multiple functors $A \to \mathcal{C}$ satisfying corresponding commutativity requirements, one for each morphism? I don't see how the above (if it even exists) accomplishes that.
I don't see how we can "replace" the morphism $a$ with the endofunctor $\alpha$. The existence of $a$ "inside of" $A$ does not seem necessary for $\alpha$ to be well-defined as a functor.
Given $\mathbf{1} \oplus \mathbf{1}$, you could just consider the (composition of) functor(s) $\mathbf{1} \oplus \mathbf{1} \to \mathbf{1} \overset{t}{\to} \mathbf{1} \oplus \mathbf{1}$ which behaves exactly the same on all of the objects and morphisms that $A$ has in common with $\mathbf{1} \oplus \mathbf{1}$.
Is there any better choice of endofunctor that could be used to "replace" $a$?
2. Inside of $Set$, consider the two element set $\{ \ast, \dagger \}$ (thus equipped with morphisms from the one-element set targeting both $\ast$ and $\dagger$ as well as with an endomorphism $f$ such that $f(\ast) = \dagger$ and $f(\dagger) = \dagger$. Then replacing $\alpha$ with $f$, $A$ with $\{\ast, \dagger\}$, $s$ with $\ast$, and $t$ with $\dagger$ (abusing notation and conflating elements of a set with maps from the one-element set to the set), this behaves the same way with respect to commutative diagrams as $A$ is supposed to in $Cat$ above.
Note on 2-category theory:
During internet searches I have found several references defining constructions similar to $A$ using $2$-categories. There is also an answer to a related Math.SE question which, when not defining $A$ in terms of itself, uses $2$-categories (via lax colimits). However, I do not think that any characterization of such a simple structure as $A$ via the framework of $2$-categories is or can be useful. I also don't see why it should be necessary.
If $2$-category theory really is necessary to axiomatize $A$ (which still seems doubtful and overkill to me), is the reason why in order to be able to capture the "internal structure" of $A$? (I.e. vis a vis the morphism $a$ and being able to distinguish $A$ from the category $\mathbf{1} \oplus \mathbf{1}$ which lacks $a$)?
And if so, can you provide a proof, or a reference proving, that $A$ can not be axiomatized using only $1$-category theory, and that $2$-category theory is truly necessary? In particular, is there some sense in which it is impossible to distinguish $\mathbf{1} \oplus \mathbf{1}$ and $A$ only using functors, and the full $2$-category structure of $Cat$ using natural transformations is somehow necessary to distinguish them?
 A: This is too long to be a comment, but is not a (good) answer, so it is community wiki.

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*Regarding your implicit question about $2$-categorical structure on $Set$, this question on MathOverflow is extremely related: Can 
be seen as a (non-trivial) 2-category? cf. e.g. the discussion in the comments about "interval objects".


*Equation (35) here defines an "arrow object" (I'm changing the notation) $A_C$
in a (strict) 2-category  for an object  such that we have an isomorphism of categories $Hom_1(,_C) \cong Hom_2(,)$ that is 2-natural in , i.e. 1-cells into $A_C$ correspond naturally to 2-cells into , or any 2-cell into  can be factored through the 2-cell between the morphisms corresponding to the identity morphism on $A_C$. Here = Two-Dimensional Models of Type Theory by Richard Garner. However it sounds like you want an "arrow object" valid for all objects in the (2-)category, a special case.


*Maybe more interesting to you are the references to Lawvere's papers in section 7 of the preprint Categories of Categories by Claudio Pisani (arXiv:math.CT/0709.0837) https://arxiv.org/pdf/0709.0837.pdf As that preprint, and the nLab article points out, one of the central tasks of that involved describing axioms for the "arrow category"/"arrow object" you describe https://ncatlab.org/nlab/show/ETCC#ETCC While there are known problems with ETCC, the axiomatization of 2/the arrow object does not seem to be one of them.


*Because having a $2$ object allows you to consider the objects of your category your category as having "internal arrows" (the same way having a terminal object 1 allows you to consider the objects of your category as having "internal points", i.e. as corresponding roughly to 0-categories a.k.a. "sets"), you should expect such axioms to be close to essentially requiring that your category can be interpreted as behaving something like a "category of categories", which of course then would have an obvious 2-categorical structure. So while maybe 2-categorical formulations aren't necessary, I would be surprised if none of them were equivalent to what you want.


*The closest I have seen on the last point (a $1$-categorical axiomatization of "arrow object" inducing a $2$-categorical structure) is this paper, A Characterization of Representable Intervals, by Michael A. Warren.
In particular, Definition 1.1. of a "cocategory object" does not require even any monoidal structure (despite the paper generally assuming if not stated otherwise that the "ambient category" $\mathcal{E}$ is bicomplete symmetric monoidal closed), it only requires that pushouts (for these special objects) exist. (Since it is supposed to be the dualization of the notion of category object, cf. internal category, which is defined using pullbacks, it shouldn't need to use any monoidal structure.)
Definition 1.6 of a "strict interval object" seems to require monoidal structure, but if you are willing to specialize to the case of "Cartesian monoidal structure", all you are really requiring is the existence of a terminal object and the existence of finite products (cf. this answer). The combination of all of these axioms for the "strict interval object" admittedly is not very elegant, and thus not obviously motivated, but that also seems to be the case for the axioms for $2$ from Lawvere's ETCC.
Then Proposition 1.18 shows how to get most of the $2$-category structure assuming the category is exponential (monoidal) closed (the internal hom is used to construct the $2$-cells and in particular to show they satisfy the axioms of a category) and finally Theorem 1.20 shows how an additional condition (I'm not sure whether this isn't always satisfied in the Cartesian case, cf. relevant nLab) guarantees the interchange law is satisfied, and thus that you get a full strict $2$-category structure.
Just to reinforce that we don't really need any monoidal structure besides Cartesian monoidal structure, Remark 1.24 of the paper says "it is an open question whether there exist examples,
aside from the trivial ones mentioned in Example 1.22, of intervals in the non-Cartesian monoidal setting for which the equivalent conditions of Theorem 1.20 hold." Moreover chapter 2 and section 2.3 of these notes (Homotopy Models of Intensional Type Theory) by the same author essentially lay out the same constructions/arguments but stated exclusively for the Cartesian closed case.
Comments/answers to a previous question seemed to argue in part that a main/defining feature of the arrow category is it being representable, and that not necessarily being related to anything about $2$-category structure. The gist of Warren's paper seems to be that this claim is only partially true -- namely the fact that the arrow category is representable, combined with all of these other categorical properties, seems to essentially force it to induce a $2$-categorical structure.
I am not sure about the extent to which requiring the original category is Cartesian (monoidal) closed is a "cop-out". At the very least, it only seems that the internal homs involving the arrow/interval object itself are required to exist for Warren's construction to work. But in any case the derivation of e.g. the definition of natural transformations (the "canonical" example of $2$-cells) using the arrow category and the Cartesian closed structure of $Cat$ is fairly standard (cf. also here as well as the several related questions about this on Math.SE).
Monoidal structure seems like possibly a "cop-out"/"circularity" to the extent that all objects of a (strict) $2$-category can be associated with a (strict) monoidal category. That being said, that might be more properly understooding as monoidal structure being required of enriching categories (cf. the comment about Kelly's notes here), but I'll be honest and admit that I don't know/understand enough about enriched category theory to understand whether the enriching category being monoidal closed is really necessary for the theory to "work" or make sense, or is just for technical convenience based on the examples most of interest. It says on nLab that "closed monoidal categories are automatically enriched over themselves via the internal hom". I suppose that doesn't necessarily imply being enriched over $Cat$ in any non-trivial way however, i.e. doesn't necessarily imply any non-trivial $2$-categorical structure by itself, e.g. $Set$ is monoidal closed, and "most" categories are enriched over $Set$, yet a priori there is no reason to believe that a given category has non-trivial $2$-category structure. Associating a monoidal category with a $2$-category with one object isn't actually really a meaningful $2$-category structure to the extent that the $1$-cells of this new one-object $2$-category correspond to the $0$-cells, not the $1$-cells, of the original monoidal $1$-category.
