# Definition:Natural Isomorphism

## Definition

Let $\mathbf C$ and $\mathbf D$ be categories.

### Covariant Functors

Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.

### Definition 1

A natural isomorphism from $F$ to $G$ is a natural transformation $\eta : F \to G$ such that for all $x\in \mathbf C$, $\eta_x : F(x) \to G(x)$ is an isomorphism.

### Definition 2

A natural isomorphism from $F$ to $G$ is an isomorphism in the functor category $\operatorname{Funct}(\mathbf C, \mathbf D)$, that is, a natural transformation $\eta : F \to G$ for which there exists a natural transformation $\xi : G \to F$ such that the compositions $\xi \circ \eta = 1_F$ and $\eta \circ \xi = 1_G$ are identity natural transformations.