Let \(\mathcal{C}\xrightarrow{F}\mathcal{D}\) be a functor. How should we define its opposite: \(\mathcal{C}^{op}\xrightarrow{F^{op}}\mathcal{D}^{op}\)
There is an isomorphism between a category and its opposite, meaning there are natural functors \(\overset{\cong}\rightarrow\) which alternate between them.
Define \(\mathcal{C}^{op}\xrightarrow{F^{op}}\mathcal{D}^{op}\) as \(F ; \overset{\cong}\rightarrow\). This is a valid functor as it is the composition of two functors.