It is common in mathematics to see definitions of the following form:

1. we begin with a certain object $A$.
2. we perform some construction depending on a choice of some parameter $\lambda\in\Lambda$ for some set $\Lambda$ which yields a new object $A\_\lambda$.
3. we show that this new object does not depend on the choice of particular $\lambda$, that is: for each pair $\lambda\_1,\lambda\_2\in\Lambda$ we have $A\_{\lambda\_1}=A\_{\lambda\_2}$.
4. at last we define $A'$ to be $A\_\lambda$ for some $\lambda\in\Lambda$.

So, to each object $A$ we have associated some object $A'$.

For example, if $r>0$ and $s\in\mathbb{R}$ are real numbers, $r^s$ is commonly seen to be defined by $r^s := \lim\_{n\to\infty}r\_n^{s\_n}$ where $\lbrace(r\_n,s\_n)\rbrace\_n$ is a sequence in $\mathbb{Q}\times\mathbb{Q}$ converging to $(r,s)$.

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> How are such definitions justified logically?
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The step that bothers me, is step 4. In order for this to really define something, shouldn't we first pick a specific element $\lambda\_A\in\Lambda$ for each object $A$ and *then* define $A':=A\_{\lambda\_A}$? Since otherwise I have a constant feeling of having to make such specific choices of $\lambda$ again and again when I should really be focusing on other things. In other words with such definitions I feel I never really know what exactly I have defined ...

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> Is there indeed some logical issue here, or am I perceiving things that aren't there?
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Wouldn't it be better to instead do something like: define $A\_\Lambda:=\lbrace A\_\lambda|\lambda\in\Lambda\rbrace$, then prove $\mathrm{card} A\_\Lambda = 1$ and then define $A':=\mathrm{UE}(A\_\lambda)$, where $\mathrm{UE}(C)$ is defined to be the unique element of $C$ if $C$ indeed has exactly one element (and $\mathrm{UE}(C)=\emptyset$ otherwise), thus eliminating the need to choose?

Thanks in advance.