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
  • …