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In the entry for lax natural transformations (https://ncatlab.org/nlab/show/lax+natural+transformation) the following axiom is given:
for each A,B, the αf are the components of a (strict) natural transformation αA,B:(αA)*∘GA,B˙→(αB)*∘FA,B
however, GA,B, FA,B, (−)*, and (−)* are left undefined.
Does any one know what their definitions are? I’m happy to extend the entry to make it more clear as well.
Thanks, Harley
The GA,B and FA,B denote the hom-functors of the given 2-functors between the pair of objects (A,B).
The (−)* and (−)* denote functors of left and right whiskering of hom-categories in a 2-category by 1-morphisms.
The αA,B then is the collection of “tin-can” diagrams to be formed by the 2-cells of the lax natural transformation.
You are completely right that, as an explanation, the text in that page is not worth the electrons that it is written on. It can only be understood by people who already understand it. If you would add explanation (and diagrams!) you’d do a great service.
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