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A matter of naming: how is the category whose objects are morphisms in a category C and whose morphisms are commutative diagrams called? and which is the nLab page dedicated to it (that’s why I need the name :) )
I generally call C→ the arrow category.
Thanks! I guess that immediately generalizes to higher categories, despite the nLab page currently does not discuss this. In particular I’m interested into some sort of category having functors from C→D as objects and commutative diagrams (including the datum of a natural transformation between the two ways of going from the upper-left to the lower-right corner) as morphisms.
You should look under category of functors and (infinity,1)-category of (infinity,1)-functors for that arrow category is Func(Δ1,C).
I’ll try to add some more cross-links to the relevant nLab entries.
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