Often in mathematics we found objects which are qualified as "being natural". The first example appears in the vector space $\mathbb R^n$, where we say that we have the "natural basis": $\{(1,0,\ldots),(0,1,\ldots),\ldots\}$. Or even the "natural" metric. The same happens in more sophisticated context: natural bundle, natural operator, natural connection,...

I know what this means, after so many years of studying maths, but only in a intuitive level: it is the simplest object of his class, and nobody has to come here to add it. In some sense, it is "included in the price". No additional data is required for it "to appear on the scene".

I also have the intuition that the "natural object" appears after fixing "something" (fixing a basis, fixing a gauge,...)

But I wonder: is there an actual definition for "natural"? I suppose that if the answer is yes, it should belong to the realm of category theory...