What is a canonical embedding and how to use it (in this context) Let ${\sim}R$ be a complete ordered field, and let $I : \mathbb Q \to {\sim} R$ be the canonical embedding.
Show that
$$ i: \mathbb R\to{\sim} R; \quad x\mapsto \sup\{ I(q) : q\in\mathbb Q; \, q\le x \} $$
defines a bijective map
How would one use the fact that is a canonical embedding in this case and does it necessarily mean that for all $q \in\mathbb Q$ that $I(q) = q$
 A: Every field has a multiplicative identity element, which here I call $\text{“}1\text{''}$ without assuming that that is the same thing as the real number $1.$ (For example, in some instances, it may be a function that is constantly equal to the real number $1.$) 
Since the field is ordered, the sequence $\underbrace{1 + \cdots + 1}_{n\text{ terms}}$ never  repeats, and in a field one can divide, so
$$
\frac{\overbrace{1 + \cdots + 1}^{m\text{ terms}}}{\underbrace{1 + \cdots + 1}_{n\text{ terms}}}
$$
exists within every ordered field. The canonical embedding takes each rational number to this image of that rational number within another ordered field (which, in the instance referred to parenthetically above, may be a function that is constantly equal to the corresponding rational number), and that much does not require completeness.
However, without completeness, you don't have the existence of the supremum that is referred to.
To prove the given map is injective is not hard; I think a proof that it's surjective may be a bit subtler. (I could address that here, but I'm not sure that's what you're asking about.)
