natural isomorphism

Definition

• A natural isomorphism \(\eta: F \Rightarrow G\) between two functors \(F, G: C \to D\) is equivalently
  • a natural transformation with two-sided inverse
  • a natural transformation each of whose components \(\eta_c: F(c) \to G(c)\) for all \(c \in C\) is an isomorphism in \(D\)
  • an isomorphism in the functor category \([C, D]\)
• In this case, we say that \(F\) and \(G\) are naturally isomorphic (\(F\) and \(G\) are isomorphic functors)

Some basic uses of isomorphic functors

• Defining the concept of the equivalence of functors
  • involves functors isomorphic to the identity functor
• Re-defining isomorphism of objects in terms of isomorphism of functors
  • ...