I believe the following nice statement is true, but I cannot find a reference or proof it myself.




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In a 2-category(i.e., bicategory), the composition of composable 2-cells is unambiguously defined.
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Where 2-cells are 2-commutative triangles (other shape will not give more information), "composable" is not easy to say in simple words but it is easy to imagine.


Special cases of this statement include vertical composition of 2-morphisms are associative and the exchange law of vertical composition and horizontal composition.


Any comments or references would be appreciated. Thank you guys!



Composition of composable 2-cells in a 2-category is unambiguously defined?
A well-known reference:



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> John Power, A 2-categorical pasting theorem, *Journal of Algebra* Volume 129, Issue 2, March 1990, Pages 439-445.
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See the [nLab page on pasting diagrams](http://ncatlab.org/nlab/show/pasting+diagram) for more. 


Power's paper just deals with *strict* 2-categories, if I remember correctly. 


**Edit:** Following Todd's comment, here's the reference to Verity's thesis, where the bicategorical case is covered. The original thesis came out in 1992, but it has recently been reprinted:



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> Dominic Verity, [Enriched categories, internal categories and change of base](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html), *Reprints in Theory and Applications of Categories* No. 20 (2011) pp 1-266.
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