Extranatural transformation

In (higher) Categorytheory an extranatural transformation is a generalization of the notion of natural transformation.

Definition

Let $F:A\times B^{op}\times B\rightarrow D$and$G:A\times C^{op}\times C\rightarrow D$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 hold:

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

$$\begin{CD}F(a,b,b^\prime)@>{F(1,1,g) }>>F(a,b,b)\\

File:Extranaturality in b.pdf

• (extranaturality in c)$\forall (h:c\rightarrow c^{\prime })\in MorC$, $\forall a\in A$, $\forall b\in B$ the folowing diagram commutes

File:Extranaturality in c.pdf