Can someone expain to me what does it mean that the exponential map between a Lie algebra and a Lie group is well defined (in a descriptive way please)?

I would have said that they are well defined because all of them assign to each element in the domain (Lie algebra) a unique elements in the codomain (Lie group)...is that correct?

For example

$\displaystyle exp: \mathfrak{so}(n)\to SO(n)$

or

$\displaystyle exp:\mathfrak{sl}(n,\mathbb{R})\to SL(n,\mathbb{R})$

and so on

thank you