# Does an adjoint pair fix a unit/counit pair?

From Ravi Vakil, Fundations of Algebraic Geometry.

I want to ask if anyone can give a hint in how to prove Execrise 1.5.B(page 43). I tried to draw the diagram for half an hour but the resulting morphisms does not seem commute at all. Nor could I prove any natural bijection must arise from some unit/counit this way.

-

You might try $\eta_A := \tau_{A,FA} (1_{FA})$.

Here is the "trick": Take any $g \colon FA \to B$. Because $\tau$ is natural in the second variable, we have commutative diagram:


Now, take $1_{FA}$ from upper left corner, and evaluate both paths in it. You get $Gg \circ \eta_A = \tau_{A,B}(g)$, as desired.

For $\epsilon$, proof is dual.

-
Thanks! I need to think about it... – Bombyx mori Feb 13 '13 at 16:44

This is the description of adjoints via unit and counit. It follows immediately from the Yoneda Lemma. Details can be found in every introduction to category theory, for example: Mac Lane, Categories for the Working Mathematician, Chapter IV, Section 1, Theorem 1.

-
Yeah, but I think it is a fun exercise. – rafaelm Feb 13 '13 at 18:29
Sure, one has to do this on its own. I just wanted to add the reference. – Martin Brandenburg Feb 13 '13 at 21:59