How do you arrive at $\eta : \text{Hom}_{C'}(Y, Y') \to \text{Hom}_{C'}(LR(Y), Y')$ from $\eta : LR \to \text{id}_{C'}$?

On page 28 of "Categories and Sheaves" it says:

$$\eta : L R \to \text{id}_{C'}$$

is a functor but then they have in a commutative diagram right below that:

$$\text{Hom}_{C'}(Y, Y') \xrightarrow{\eta_Y} \text{Hom}_{C'}(LR(Y), Y')$$

How does that make sense?

• do you mean it is a natural transformation between functors? because what you have written there looks an awful lot like a counit! – Enkidu Oct 30 '18 at 8:21

You have the morphism $$\eta^Y : LR(Y) \to Y$$ and you apply the contravariant functor $$\mathrm{Hom}_{C'}(-,Y')$$. This yields the desired morphism $$\text{Hom}_{C'}(Y, Y') \xrightarrow{\eta_Y} \text{Hom}_{C'}(LR(Y), Y').$$
• Shouldn't we put a $*$ next to $\eta_Y$: $\eta_Y^*$ since taking the domain-changing hom functor results in composition with $\eta_Y$ on the right? – BananaCats Category Theory App Nov 1 '18 at 0:04
• @RollupandsmokeAdjoint : I don't know the notations from your book, but I agree with you, we may write $\eta_Y := (\eta^Y)^*$, or write $\eta_Y :LR(Y) \to Y$ and $\eta_Y^* : \text{Hom}_{C'}(Y, Y') \to \text{Hom}_{C'}(LR(Y), Y')$. – Watson Nov 1 '18 at 8:31