I am learning about adjoint functors, so far I have learned a lot about equivalent definitions of adjoint functors.
The one thing I still can't understand very clearly is the definition of adjoint functors via triangle identities.
This definition says:
Given two functors $F:D \to C$, $G:C \to D$, an adjunction $F \dashv G$ is given by two natural transformations
$η:1_{D} \to GF$
$ϵ:FG\to 1_{C}$
such that the following two triangles commute
i.e.
$ϵF ∘ Fη = 1_{F}$
$Gϵ ∘ ηG = 1_{G}$
Note: Since the triangle on the left is in the functor category $[D, C]$, and the triangle on the right is in the functor category $[C, D]$, so triangle identities may essentially be re-expressed as the language of morphisms (left/right) inverse in the functors categories.
So can I restate the adjoint functors as follows?
Given two functors $F:D \to C$, $G:C \to D$, an adjunction between F and G ($F \dashv G$ or $G \dashv F$) is given by four natural transformations
$η_{1} : F \to FGF$
$ϵ_{1} : FGF \to F$
$η_{2} : G \to GFG$
$ϵ_{2} : GFG \to G$
such that the following two triangles commute,
i.e.
$1_{F} = ϵ_{1} ∘ η_{1}$
$1_{G} = ϵ_{2} ∘ η_{2}$
This definition seems more intuitive than the original definition, because we can think of $η_{1}$ as the right inverse of $ϵ_{1}$ and $η_{2}$ as the right inverse of $ϵ_{2}$.
What is the difference between these two definitions?
Is there a relationship between them?
Very thanks.