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}