Extranatural Transformation

Let F : A × B op × B → D : 𝐹 → 𝐴 superscript 𝐵 op 𝐵 𝐷 F:A\times B^{\mathrm{op}}\times B\to D , G : A × B op × B → D : 𝐺 → 𝐴 superscript 𝐵 op 𝐵 𝐷 G:A\times B^{\mathrm{op}}\times B\to D be a pair of functors. An extranatural transformation α : F → G : 𝛼 → 𝐹 𝐺 \alpha:F\to G consists of a family of maps α a , b , c : F ⁢ ( a , b , b ) → G ⁢ ( a , c , c ) : subscript 𝛼 𝑎 𝑏 𝑐 → 𝐹 𝑎 𝑏 𝑏 𝐺 𝑎 𝑐 𝑐 \alpha_{a,b,c}:F(a,b,b)\to G(a,c,c) that satisfy the following conditions:

F⁢(a,b,b)𝐹𝑎𝑏𝑏{{F(a,b,b)}}G⁢(a,c,c)𝐺𝑎𝑐𝑐{{G(a,c,c)}}F⁢(a′,b,b)𝐹superscript𝑎′𝑏𝑏{{F(a^{\prime},b,b)}}G⁢(a′,c,c)𝐺superscript𝑎′𝑐𝑐{{G(a^{\prime},c,c)}}αa′,b,csubscript𝛼superscript𝑎′𝑏𝑐\scriptstyle{\alpha_{a^{\prime},b,c}}F⁢(f,i⁢d,i⁢d)𝐹𝑓𝑖𝑑𝑖𝑑\scriptstyle{F(f,id,id)}G⁢(f,i⁢d,i⁢d)𝐺𝑓𝑖𝑑𝑖𝑑\scriptstyle{G(f,id,id)}αa′,b,csubscript𝛼superscript𝑎′𝑏𝑐\scriptstyle{\alpha_{a^{\prime},b,c}}

F⁢(a,b′,b)𝐹𝑎superscript𝑏′𝑏{{F(a,b^{\prime},b)}}F⁢(a,b′,b′)𝐹𝑎superscript𝑏′superscript𝑏′{{F(a,b^{\prime},b^{\prime})}}F⁢(a,b,b)𝐹𝑎𝑏𝑏{{F(a,b,b)}}G⁢(a′,c,c)𝐺superscript𝑎′𝑐𝑐{{G(a^{\prime},c,c)}}F⁢(i⁢d,i⁢d,g)𝐹𝑖𝑑𝑖𝑑𝑔\scriptstyle{F(id,id,g)}F⁢(i⁢d,g,i⁢d)𝐹𝑖𝑑𝑔𝑖𝑑\scriptstyle{F(id,g,id)}αa,b′,csubscript𝛼𝑎superscript𝑏′𝑐\scriptstyle{\alpha_{a,b^{\prime},c}}αa,b,csubscript𝛼𝑎𝑏𝑐\scriptstyle{\alpha_{a,b,c}}

F⁢(a,b,b)𝐹𝑎𝑏𝑏{{F(a,b,b)}}G⁢(a,c′,c′)𝐺𝑎superscript𝑐′superscript𝑐′{{G(a,c^{\prime},c^{\prime})}}G⁢(a,c,c)𝐺𝑎𝑐𝑐{{G(a,c,c)}}G⁢(a,c,c′)𝐺𝑎𝑐superscript𝑐′{{G(a,c,c^{\prime})}}αa,b,c′subscript𝛼𝑎𝑏superscript𝑐′\scriptstyle{\alpha_{a,b,c^{\prime}}}αa,b,csubscript𝛼𝑎𝑏𝑐\scriptstyle{\alpha_{a,b,c}}G⁢(i⁢d,h,i⁢d)𝐺𝑖𝑑ℎ𝑖𝑑\scriptstyle{G(id,h,id)}G⁢(i⁢d,i⁢d,h)𝐺𝑖𝑑𝑖𝑑ℎ\scriptstyle{G(id,id,h)}