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The question is in the title, really.

I understand the answer in a general sense: Morphisms map objects, and functors map both objects and morphisms. So an endofunctor would map morphisms as well as objects. But that's not an intuitively satisfying answer. It seems like "part" of the endofunctor is just a morphism under a different name. I'm trying to understand this at a deeper level, and the texts I have don't really cover this question, per se.

Examples might be helpful, or more clarification on the ways they are different beyond the simple version I give above. I'm not sure how to clarify what I'm after yet, but feel free to try and pin me down to specifics in comments if it helps...

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    I don't really understand the question. Specializing to the case of a category with one object and all morphisms invertible (a group in disguise) we get the question "what's the difference between an endomorphism of a group and an element of a group?" and I don't know what to say to this except "one of these things is not like the other..."2011-07-07
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    @Qiaochu, I suspect the misunderstanding is mine, rather than yours, but I'm trying to pin it down to specifics. A morphism maps an object in a category to another object in the category, right? A functor maps both objects and morphisms in one category to objects and morphisms in another category, right? An endofunctor is a functor which maps from one category back onto itself, right? So one of the things an endofunctor does is to map objects from a category onto other objects in the category, which is what a morphism does. It *feels* like there is overlap, but as you say, they're different.2011-07-07
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    A morphism is an arrow from some _specific_ object to some _specific_ other object. An endofunctor contains the data of a function from the _collection_ of objects in some category to itself. I think you should work through some examples more closely to see what the difference between these two notions is; they're on two different categorical levels. One way to say this is that a functor is a morphism between two categories in the "category of categories."2011-07-07
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    @Qiaochu, yes, as I said, an example or two is really what I'm looking for. Your comment above is probably half of the answer, a good reference to an example or two is probably the other half.2011-07-07

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