Let $C,D$ be categories and $F:C\to D,G:D\to C$ be such that $F$ is a left adjoint of $G$. Prove that $F$ is fully faithful iff the unit is an isomorphism.
(This is an exercise from the book by T. Leinster)
I think I can do one direction:
$\impliedby$: If we let $\eta$,$e$ be unit/counit then as $\eta:\text{id}_C \implies GF$ is an isomorphism, it follows that the composite
$$
\text{Hom}(x,y)\to\text{Hom}(F(x),F(y))\to\text{Hom}(GF(x),GF(y))
$$
for all $x,y\in C$ is an isomorphism, which implies that $F$ is faithful. But $F$ must be full since the composite
$$
G=G\circ \text{id}_D \implies GFG \implies G
$$
is an identity transformation (I'm rather skeptical about this).
But I don't have any idea regarding reverse direction, in which I have to show that the arrow $x\to GF(x)$ has an inverse for all $x\in C$.
Best Answer
$F$ 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.