Extranatural transformation

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In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.


Let F:A\times B^\mathrm{op}\times B\rightarrow D and  G:A\times C^\mathrm{op}\times C\rightarrow D two functors of categories. A family \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c) is said to be natural in a and extranatural in b and c if the following holds:

F(a,b,b') & \xrightarrow{F(1,1,g)} & F(a,b,b) \\
_{F(1,g,1)}|\qquad & & _{\eta(a,b,c)}|\qquad \\
F(a,b',b') & \xrightarrow{\eta(a,b',c)} & G(a,c,c)
  • (extranaturality in c) \forall (h:c\rightarrow c^\prime)\in \mathrm{Mor}\, C, \forall a\in A, \forall b\in B the following diagram commutes
F(a,b,b) & \xrightarrow{\eta(a,b,c)} & G(a,c,c) \\
_{\eta(a,b,c')}|\qquad & & _{G(1,h,1)}|\qquad \\
G(a,c',c') & \xrightarrow{G(1,1,h)} & G(a,c',c)