Consider the following definition:
Let $X$ be a set and $e : X \mapsto A$ a mapping to a boolean algebra $A.$ We say that $A$ is free over $X$ (with respect to $e$) if for every mapping $f:X \mapsto B$ for a boolean algebra $B$ there is precisely one homomorphism $\overline{f}:A \mapsto B$ such that $\overline{f} \circ e = f.$
I would eventually like to prove some statements involving free boolean algebras but I am stuck in understanding this definition.
I understand it formally but don't see its intuitive meaning. Something simply doesn't click.
How should one think about this definition?
For example, how should one try to prove that if $A$ is free over $X$ with respect to $e:X \mapsto A$ then $e$ is injective? Is it correct to simply say that if $e$ is not injective then so isn't $\overline{f}\circ e$ but since $f$ is arbitrary it can as well be an injective function?
What about the fact that any free algebra over a set of cardinality $n$ has precisely $2^{2^{n}}$ elements? Is there a direct way (without referring to the Lindenbaum algebra) to prove this claim directly from the definition?
(PS. I am no category theorist the arrow notation just scares me :)
\to
in TeX). The "$\mapsto$" arrow\mapsto
is for "the function we're speaking of maps this particular element to that one". $\endgroup$