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?

  • $\begingroup$ do you mean it is a natural transformation between functors? because what you have written there looks an awful lot like a counit! $\endgroup$ – 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 $$ \text{Hom}_{C'}(Y, Y') \xrightarrow{\eta_Y} \text{Hom}_{C'}(LR(Y), Y'). $$

  • $\begingroup$ 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? $\endgroup$ – BananaCats Category Theory App Nov 1 '18 at 0:04
  • 1
    $\begingroup$ @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 $\eta_Y^* : \text{Hom}_{C'}(Y, Y') \to \text{Hom}_{C'}(LR(Y), Y')$. $\endgroup$ – Watson Nov 1 '18 at 8:31

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