Two Questions on Peter Smith's Representation of Category Theory Background
Recently I was going through Peter Smith's Category Theory : A Gentle Introduction. There 
he writes (in section 1.1), 

Definition 1. A category $\mathscr{C}$ comprises two kinds of things:
(1) $\mathscr{C}$ -objects (which we will typically notate by ‘$A$’, ‘$B$’, ‘$C$’, $\ldots$).
(2) $\mathscr{C}$ -arrows (which we typically notate by ‘$f$’, ‘$g$’, ‘$h$’, $\ldots$ ).
These objects and arrows are governed by the following axioms:
Sources and targets For each arrow $f$, there are unique associated objects 
  $src(f)$ and $tar (f)$, respectively the source and target of $f$, not necessarily distinct.
We write ‘$f : A → B$’ or ‘$A\xrightarrow{f}B$’ to notate that $f$ is an arrow with $src(f) = A$ and $tar (f) = B$.
Composition For any two arrows $f : A → B, g : B → C$, where $src(g) = tar (f)$, there exists an arrow $g \circ f : A → C$, ‘$g$ following $f$’, which we call the composite of $f$ with $g$.
Identity arrows Given any object $A$, there is an arrow $1_A : A → A$ called the identity arrow on $A$.
$\ldots$
Associativity of composition. For any $f : A → B, g : B → C, h: C → D$, we
  have $h \circ (g \circ f) = (h \circ g) \circ f$.
  Identity arrows behave as identities. For any $f : A → B$ we have $f \circ 1_A = f = 1_B ◦ f$.

Now after writing this he continues with six remarks and in the fourth of them he writes,

In keeping with the functional paradigm, the source and target of an arrow
  are very often called, respectively, the ‘domain’ and ‘codomain’ of the
  arrow. But that usage has the potential to mislead when arrows aren’t
  (the right kind of) functions, which is again why I prefer our terminology.

Very shortly there after he proves Theorem 1 which I am quoting verbatim,

Theorem 1. Identity arrows on a given object are unique; and the identity arrows on distinct objects are distinct.
Proof. Suppose $A$ has identity arrows $1_A$ and $1'_A$. Then applying the identity axioms for each, $1_A = 1_A \circ 1'_A = 1'_A\circ 1_A$.
For the second part, we simply note that $A\ne B$ entails $src(1_A) \ne src(1_B)$ which entails $1_A \ne 1_B$.

Questions


*

*If the target of an arrow is analogous concept of codomain for a function then how can it be unique (as is said in the first axiom)?

*In the proof of the second part of Theorem 1, I understand that $A\ne B$ implies that $src(1_A)\ne src(1_B)$ but how from $src(1_A)\ne src(1_B)$ it follows that $1_A\ne 1_B$? For more details on this see I, II, III, IV (in this order), especially IV. 
 A: 1. There are slightly different conceptions of function. What you're assuming is the formalization usually presented in the context of set theory, where a function is simply a set of ordered pairs satisfying certain conditions. In that case it is unambiguous what the domain and range of the function is, but the "codomain" can be taken to be any (proper or not) superset of the range.
However, in many other places, a "function" is considered to be something that inherently knows what its codomain is. In that case we would model it set-theoretically as a triple $(D,R,C)$ where $D$ and $C$ are sets and $R$ is a subset of $D\times C$ that satisfies certain conditions. In this case the codomain of $f=(D,R,C)$ is by definition $C$, and therefore a function can have only one codomain.
This is not as strange as it may sound as first if you're used to the former variant. Consider, for example, that even introductory courses happily talks about whether a function is "surjective" or not -- and in order to make sense of that at all, it needs to be the kind of "function" that knows what its codomain is.
It is the latter conception of function that is analogous to Peter Smith's presentation of arrows in a category.
(There are other presentations of category theory where an arrow doesn't necessarily know, by itself, what its domain and codomain are. In those formalizations you need to keep track of which hom-sets the things you are speaking about come from yourself).
2. If you accept that an arrow has only one source (that is, $\rm src$ is an inherent property of every arrow), then it is impossible for $1_A$ to equal $1_B$ yet have different sources -- because two things (such as arrows, or anything in mathematics) that are equal are the same thing and therefore cannot have different properties depending on what you call them.
A: I needn't add anything of substance to Henning Makholm's answer.
Except perhaps to note that if you'd read on all the way as far as p. 7 of my notes, your question would have been answered!
I there explicitly note how in the category Set, the arrows aren't functions-as-usually-defined-in-set-theory, but functions-with-an-assigned-codomain.
Moral: always a good idea to look at an author's examples given to illustrate an abstract definition in order to check your understanding.
A: 1)  Apparently the target of $f:A\to B $ is $B $...  It is part of the definition of the arrow $f $.
2)  Equal arrows would have to have the same source...
A: In usual treatments of set theory one defines a function $f$
from $A$ to $B$ as the set $\{(a,f(a)):a\in A\}$. Then for instance, the identity map $\Bbb Q\to \Bbb Q$ and the inclusion map
 $\Bbb Q\to \Bbb R$, are represented by the same set. Are these really
the same function? In category theory, these are regarded as different
morphisms in the category of sets, as each arrow determines it target.
One way to model this in set theory would be to identify a function
$f:A\to B$ as the ordered triple $(A,B,\{(a,f(a)):a\in A\})$.
A: I would like to point out that Maarten Fokkinga's 1992 introduction to category theory, A Gentle Introduction to Category Theory: the calculational approach, addresses this head-on by defining a notion of "pre-category" where hom-sets are not required to be disjoint. This, arguably, more naturally fits many of the examples we have. In particular, the set-theoretic definition of function doesn't uniquely determine the codomain (as opposed to the range) and thus, sets and functions more naturally form a pre-category rather than a category. Of course, it's easy to turn a pre-category into a category.
This introduction overall is rather unique in many ways. While I don't personally recommend doing category in the style espoused, I do recommend exposing yourself to this style to have a different perspective.
Finally, the term "precategory" is used for two other closely related but quite different notions. Indeed, I suspect this paper (or possibly some other papers by the same or related authors) is the only place "pre-category" is used in the above sense.
