Let $C,D$ be categories and $F:C\to D$ and $G:D\to C$ be adjoint functors. Then $F$ is fully faithful iff the unit is an isomorphism?
Solution 1:
$F$ is fully faithful iff the map $\text{Hom}(x, y) \to \text{Hom}(Fx, Fy)$ is an isomorphism. By adjunction, the map $\text{Hom}(Fx, Fy) \to \text{Hom}(x, GFy)$ is always an isomorphism. Now use the Yoneda lemma; this proves both directions simultaneously.
Solution 2:
It's actually possible to prove something a bit more general :
- $F$ is faithful if and only if every component of $\eta $ is a monomorphism.
- $F$ is full if and only if every component of $\eta$ is a split epimorphism.
The first statement holds because for every objects $X,Y$ and every arrows $u,v:X\to Y$ of $C$, $$\eta_Y\circ u=\eta_Y\circ v\Longleftrightarrow F(u)=F(v)$$ (because the two equalities correspond to one another through the natural bijection $\operatorname{Hom}_C(X,GFY)\simeq \operatorname{Hom}_D(FX,FY)$).
For the second statement, first assume that every $\eta_X$ is a split epimorphism, with some section $s_X$. Take an arrow $g:FX\to FY$, then $f=s_Y\circ G(g)\circ \eta_X$ is an arrow $X\to Y$, and $$GF(f)\circ \eta_X=\eta_Y\circ f = G(g)\circ \eta_X,$$which shows that $F(f)=g$. Assume now that $F$ is full; then for every $X$ there must be some arrow $s_X:GFX\to X$ such that $F(s_X)=\epsilon_{FX}:FGFX\to FX$. Now $$\epsilon_{FX} \circ F(\eta_X\circ s_X)=\epsilon_{FX} \circ F(\eta_X)\circ \epsilon_{FX}=\epsilon_{FX}=\epsilon_{FX}\circ F(id_{GFX}),$$hence $\eta_X\circ s_X=id_{GFX}$.