Extranatural transformation

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

Definition

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:

• $\eta (-,b,c)$ is a natural transformation (in the usual sense).
• (extranaturality in b) $\forall (g:b\rightarrow b^{\prime })\in \mathrm {Mor} \,B$, $\forall a\in A$, $\forall c\in C$ the following diagram commutes

${\begin{matrix}F(a,b',b)&{\xrightarrow {F(1,1,g)}}&F(a,b',b')\\_{F(1,g,1)}|\qquad &&_{\eta (a,b',c)}|\qquad \\F(a,b,b)&{\xrightarrow {\eta (a,b,c)}}&G(a,c,c)\end{matrix}}$

• (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

${\begin{matrix}F(a,b,b)&{\xrightarrow {\eta (a,b,c')}}&G(a,c',c')\\_{\eta (a,b,c)}|\qquad &&_{G(1,h,1)}|\qquad \\G(a,c,c)&{\xrightarrow {G(1,1,h)}}&G(a,c,c')\end{matrix}}$