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I thought that this was dense, but looking at the page dense functor then I’m wrong.
To be clear, I have a functor $F \colon C \to D$ with the property that every object in $D$ is isomorphic to one in the image of $C$. In my case, $F$ happens also to be an inclusion of a subcategory. What’s the name for this?
I think that the first property makes your functor an essentially surjective functor (and the second makes it faithful).
That looks right! Thanks.
So “essentially surjective subcategory”?
That sounds a bit yucky, but fortunately I have the functor explicitly so I can say “essentially surjective functor”.
How about essentially wide subcategory?
Looks good to me! Thanks. So we have essentially surjective functor for the functor, and essentially wide subcategory for the subcategory. Great.
I added the terminology for the subcategory for the page on the functor (the reverse was already there).
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