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 $$ G:A\times C^\mathrm{op}\times C\rightarrow D $$ be 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)}\downarrow\qquad & & _{\eta(a,b',c)}\downarrow\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)}\downarrow\qquad & & _{G(1,h,1)}\downarrow\qquad \\ G(a,c,c) & \xrightarrow{G(1,1,h)} & G(a,c,c') \end{matrix}$$

Properties
Extranatural transformations can be used to define wedges and thereby ends (dually co-wedges and co-ends), by setting $$F$$ (dually $$G$$) constant.

Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case.