# Definition of adjoint functors in Leinster's "Basic Category Theory"

Here is how the definition of adjoint functor reads on page 41 of Leinster's "Basic Category Theory":

I understood this part thanks to Definition of Adjunction in Category Theory. It clarifies that the adjunction is a natural isomorphism between certain functors. However, as Remark 2.1.2 on the book seem to suggest, Leinster is postponing this interpretation to Chapter 4 and transcribing the definition of natural isomorphism for these functors:

Why is he doing so?

• Probably he planned to introduce natural transformations only in chapter 4, for some reason.. Commented Jul 24, 2019 at 17:12
• @Berci but natural transformations are introduced in chapter 1... Commented Jul 24, 2019 at 17:13
• Ahh, indeed? Then I see no reason.. Commented Jul 24, 2019 at 17:14

Actually the isomorphisms are a little bit more than natural transformations. Suppose the isomorphisms are denoted by $$\eta_{A,B}:\mathscr{B}(F(A),B)\to\mathscr{A}(A,G(B))$$, then for fixed $$A\in\mathscr{A}$$, $$\eta_{A,-}$$ is a transformation $$\mathscr{B}(F(A),-)\Rightarrow\mathscr{A}(A,G(-))$$, and similarly for fixed $$B\in\mathscr{B}$$, $$\eta_{-,B}$$ is a transformation $$\mathscr{B}(F(-),B)\Rightarrow\mathscr{A}(-,G(B))$$. The "wired" statements are just saying these.
These conditions make $$\eta_{-,-}$$ a natural transformation between bifunctors.