Arrow category, are source and target well defined? Using definition here.
I am trying to learn category theory and have just been introduced to arrow category, however in the definition I noticed a (possible) discrepancy in the well definedness of the category. 
Let $C = \mathcal{Set}$ and consider $C^{\downarrow}$.
In that case both: 
$\require{AMScd}$
A =
\begin{CD}
    \mathbb{R} \\
    @VVV 1_{\mathbb{R}}\\
    \mathbb{R}
\end{CD}
,B = 
\begin{CD}
    \mathbb{R} \\
    @VVV 2\cdot1_{\mathbb{R}}\\
    \mathbb{R}
\end{CD} 
$\in {C^{\downarrow}}_{Obj}$
Also noting That the following commute:
$\require{AMScd}$
\begin{CD}
    \mathbb{R} @>1_{\mathbb{R}}>> \mathbb{R}\\
    @V 1_{\mathbb{R}} V V @VV 1_\mathbb{R} V\\
    \mathbb{R} @>>1_{\mathbb{R}}> \mathbb{R}
\end{CD},
$\require{AMScd}$
\begin{CD}
    \mathbb{R} @>1_{\mathbb{R}}>> \mathbb{R}\\
    @V 2\cdot1_{\mathbb{R}} V V @VV 2\cdot1_\mathbb{R} V\\
    \mathbb{R} @>>1_{\mathbb{R}}> \mathbb{R}
\end{CD} 
.
However that would mean $(1_{\mathbb{R}},1_{\mathbb{R}}):A\to A$ and $(1_{\mathbb{R}},1_{\mathbb{R}}):B\to B$. But as $A \neq B$ it follows that $(1_{\mathbb{R}},1_{\mathbb{R}})$ has two sources and two targets, this is not well defined. If anyone could help shed some light into my misunderstanding it would be greatly appricted.
 A: This is a practical notation, so what we want is a way to interpret it in a systematic fashion. There are at least three methods. To fix notation, I will speak in terms of the diagram
$$\require{AMScd} \begin{CD}
W @>f>> X 
\\ @VV aV @VV bV
\\ Y @>g>> Z
\end{CD} $$

One can view $(f, g)$ as merely being shorthand for an unstated mathematical object that correctly remembers the domain and codomain as well, such as the quadruple $(a,f,g,b)$.
This is probably the most traditional approach.

One can devise a notation for categories in which one has a collection of labels: in this category, $(f,g)$ would be an example of such a label.
The labels are not arrows themselves. Instead, each arrow of the category is assigned a label.
I've seen this approach taken in Categories, Allegories, which calls the labels "proto-morphisms" and also gives an axiomatization of categories in terms of objects and proto-morphisms rather than in terms of objects and arrows. See this nLab entry for more details.
This approach is neat because it aims to describe the actual practice of how people work with categories, rather than copping out and saying the actual practice is shorthand as in the above section.
Unfortunately, I don't think I've seen it (explicitly) used anywhere else.

A third approach, based on dependent type theory, is that the arrows of a category should be organized as a family $\hom_\mathcal{C}(X,Y)$ of homsets, rather than as a class $\mathrm{Mor}(\mathcal{C})$ of all morphisms.
In this organization, it never makes sense to talk about a "free" arrow — you only talk about arrows in the context of being an arrow between two specific objects.
Thus, there is no ambiguity in $(1_\mathbb{R}, 1_\mathbb{R})$, because you wouldn't talk about such an arrow of $\mathrm{Set}^{\downarrow}$ without having previously stated which objects this should be an arrow between.
When defining a category in terms of homsets, the homsets for different pairs of objects are allowed to overlap... but you really shouldn't be asking questions 
whose answer depends on whether or not they overlap... and if you insist on asking such questions you'd probably prefer to switch to an isomorphic category where they do not overlap.
A: You seem to be making the assumption that the pair arrows $(f_1,f_2)$ in
\begin{CD}
    a_0 @>f_0>> b_0\\
    @V a V V @VV b V\\
    a_1@>>f_1> b_1
\end{CD} 
is the morphism $a\to b$. But this is not true! The morphism $a\to b$ is literally defined to be the entire commutative diagram above. So in your examples you give, the two diagrams give two separate morphisms $1_{\Bbb R}\to 1_{\Bbb R}$ and $2\cdot 1_{\Bbb R}\to2\cdot1_{\Bbb R}$, and there is no problem.
