Editing Functor (section)

=== Opposite functor ===
Every functor <math>F \colon  C\to D</math> induces the '''opposite functor''' <math>F^\mathrm{op} \colon  C^\mathrm{op}\to D^\mathrm{op}</math>, where <math>C^\mathrm{op}</math> and <math>D^\mathrm{op}</math> are the [[opposite category|opposite categories]] to <math>C</math> and <math>D</math>.<ref>{{citation|first1=Saunders|last1=Mac Lane|author-link1=Saunders Mac Lane|first2=Ieke|last2=Moerdijk|author-link2=Ieke Moerdijk|title=Sheaves in geometry and logic: a first introduction to topos theory|publisher=Springer|year=1992|isbn=978-0-387-97710-2}}</ref> By definition, <math>F^\mathrm{op}</math> maps objects and morphisms in the identical way as does <math>F</math>. Since <math>C^\mathrm{op}</math> does not coincide with <math>C</math> as a category, and similarly for <math>D</math>, <math>F^\mathrm{op}</math> is distinguished from <math>F</math>. For example, when composing <math>F \colon  C_0\to C_1</math> with <math>G \colon  C_1^\mathrm{op}\to C_2</math>, one should use either <math>G\circ F^\mathrm{op}</math> or <math>G^\mathrm{op}\circ F</math>. Note that, following the property of [[opposite category]], <math>\left(F^\mathrm{op}\right)^\mathrm{op} = F</math>.