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

Question

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?

Answer 1

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