# Full or faithful adjoints

(draft)

## Summary

Properties of adjoints and (co)units are linked as follows.

| counit componentwise | right adjoint|
|:---:|:---:|
| epi | faithful |
| split mono | full |

In particular, a fully faithful right-adjoint is a reflective subcategory. The subcategory is the category of algebras for the monad induced by the adjunction.

| unit componentwise| left adjoint|
|:---:|:---:|
| mono | faithful |
| split epi | full |

In particular, a fully faithful left-adjoint is a coreflective subcategory. The subcategory is the category of coalgebras for the comonad induced by the adjunction.

## The Theorem

Let $F:X\to A$, $G:A\to X$, $F\dashv G$, $\eta_x:x\to GFx$, $\epsilon_a:FGa\to a$.


Mac Lane, page 88:


This is proved using:


## The Proof

For variety, we prove the dual of the lemma and the theorem.

**Lemma:** Let $f^\ast:X(-,y) \to X(-,z)$ given by Yoneda for $f:y\to z$. Then $f^\ast$ is componentwise injective iff $f$ is mono and $f^\ast$ is componentwise onto iff $f$ is split epi.

*Proof:* This is straightforward from the respective definitions.

To prove the dual of the theorem, consider
$$X(x,y)\stackrel {F_{x,y}} \longrightarrow
A(Fx,Fy) \stackrel \cong \longrightarrow
X(x,UFy)$$

which is the transformation natural in $x$ determined (via Yoneda) by the unit $\eta_y:y\to UFy$.

The lemma tells us that the natural transformation, hence $F_{x,y}$, is componentwise injective iff $\eta_y$ is mono and is componentwise surjective iff $\eta_y$ is split epi.

We have shown that $F$ is faithful iff the unit is componentwise mono and full iff the unit is componentwise split epi.

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