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There are some really good discussions of the meaning/ambiguity of the term "natural" here and here.

One thing I didn't quite get from those answers: when we say the Snake Lemma, for instance, is "natural", shouldn't we really say it is "functorial"? In other words, it doesn't just associate with each diagram a long exact sequence, it takes maps between diagrams to maps between long exact sequences?

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Eric Auld
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    Natural sometimes means functorial and sometimes means something involving natural transformations. It's unfortunate but that's how people use the term. – Qiaochu Yuan Dec 10 '15 at 19:40
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    How I like to think of it is that 'naturality' is when an object mapping can be naturally extended (by an arrow mapping) to a functor. A natural transformation can be seen as a special case when the target category is an arrow category. – Berci Dec 10 '15 at 20:59

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