# Defining adjoint functors: What does “natural bijection” mean?

Take the following definition of adjunction from the nlab

This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for every $c$ and $d$ the hom-sets are naturally isomorphic. At his point it does SCRREEETCH, because, as a beginner, learning category theory, natural isomorphisms were only defined between functors as a collection of morphisms in the target category and not *between collections of morphisms lying in different categories.

I know that there are other equivalent definitions of adjunctions, but I don't have the time to go through them to see how to make sense of this definition. I just want a clean explanation how to understand this definition.

• $\text{Hom}_D(L(c),d)$ and $\text{Hom}_C(c,R(d))$ are both sets, so they're objects of the category $\text{Set}$, which is the target category where the natural isomorphism is taking place. It's not that the Hom sets are naturally isomorphic - there's just a bijection between them, and these bijections cohere to a natural ismorphism between the Hom functors. – Alex Kruckman Mar 14 '16 at 22:16
• The second sentence is just repeating what the first sentence is saying in more basic terms. If you unfold the definition of natural isomorphism you will see this. – Zhen Lin Mar 14 '16 at 22:17
• Why does the nlab page say "natural isomorphism" for what has traditionally been called a "natural equivalence"? – Rob Arthan Mar 14 '16 at 22:25
• @Rob Arthan natural isomorphism is very standard, to my mind moreso than natural equivalence. – Kevin Arlin Mar 14 '16 at 23:13
• @KevinCarlson: is it? I use Mac Lane's "Categories for the Working Mathematician" as a primary source for basic category theory notions. If "natural isomorphism" has taken over from "natural equivalence", then do you call "natural transformations" "natural homomorphisms"? – Rob Arthan Mar 14 '16 at 23:23

$\text{Hom}_D(L(c),d)$ and $\text{Hom}_C(c,R(d))$ are both sets, i.e. they're objects of the category $\mathsf{Set}$, which is the target category where the natural isomorphism is taking place.
It's not that the $\text{Hom}$ sets are naturally isomorphic - there's just a bijection between them, and these bijections cohere to a natural ismorphism between the $\text{Hom}$ functors.