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From what I’ve absorbed about functor categories and size issues, it seems that (perhaps with a suitable choice axiom), the category of functors between two fixed categories is only guaranteed to be locally small if the source category is essentially small.
If true, I feel it should be noted somewhere. If false, ditto since it’s a fairly obvious conclusion to draw. Unfortunately, I couldn’t find this anywhere on the nLab. I looked at small category, locally small category, functor category, and even at cat.
I’m quite happy to put it in - indeed, by Urs’ Law then it’s my job to do so - but I’d like confirmation that it is a correct statement before doing so.
I added a bit on size issues to functor category.
Thanks! I knew someone else would be able to do it better than me: what you wrote is what I meant to say in my first post in this discussion but I missed off a few “locally small”s.
I added a partial converse to the last statement at functor category.
That’s a great converse! I have commented on it in another thread.
Well, the relevant among the old entries is small presheaf.
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