Category Theory and Polar Arithmetic

F\textup{ and }G\textup{ are covariant functors between the categories }C\textup{ and }D\textup{,}\\\textup{then a natural transformation }\eta\textup{ from }F\textup{ to }G\textup{ maps every object }x\textup{ in }C\\\textup{a morphism }\eta{_x}:F(x)\rightarrow G(x)\textup{ in }D\textup{ such that }\forall \textup{ morphism }f:x\rightarrow y\\\textup{in }C\textup{, }\eta{_y}\circ F(f)=G(f)\circ \eta{_x}