I am sorry if this is an obvious question.
I have a curious situation, in a software design, where it's supposed that there may exist a space $N$ (for normalised data) and a function $e: N \to K$ ($K$ another space different from $N$), such that for each function $h$ and space $L$ where $h: L \to K$, there exists a function $i: L \to N$, verifying $h = e\circ i$.
By intuition, I think this is not possible, or it must be some additional conditions on the spaces $L$, $N$ and $K$.
Is this a known problem in category theory ? If yes, are there possible solutions knowing that $e= id$ and $N = K$ is not an option ?
Thanks in advance and sorry if this is a badly formulated question
Your condition means precisely that $e$ is a retraction, i.e. that $e$ admits a right inverse, i.e. that there exists a morphism $f \colon K \to N$ with $e \circ f = \operatorname{id}_K$.
To see that such an $f$ exists under the given conditions we take $L = K$ and $h = \operatorname{id}_K$ and then take for $f$ the resulting morphism $i$. If on the other hand $e$ admits a right inverse $f$ then we can always get the required morphism $i$ as $i := f \circ h$.
In the category of sets, i.e. if $K, L, N, \dotsc$ are sets and $e, h, i, \dotsc$ maps then this means precisely that $e$ is surjective.