Characterizing Representable / Hom Functors via Universal Property

I have been trying to think of a way to characterize the hom functors via universal property. I could not find any such thing elsewhere online. So I came up with a property, inspired by Yoneda lemma, and a proof that any functor satisfying this property must be representable: For a category $C$ , let $R\in Set^>$ such that $Hom_>(R,F)\cong Fc$ , natural in $F$ . Then $R\cong C(-,c)$ . This would lend the universal arrow $(R, r : 1 \to Rc)$ from $1=\$ to the functor that sends $F\mapsto Fc$ and $(\alpha: F\Rightarrow G)\mapsto (\alpha)_c$ . The element $r\in Rc$ is defined in the proof, sorry for putting that before hand. But it is basically the analog of $1_c\in C(c,c)$ . I will include a quick proof that any such $R$ satisfies $R\cong C(-,c)$ . But, before that, here is my question: is there something like this already out there, i.e., a characterization of representable functors via universal properties? Does this suffice to define hom functors? Of course this will not give us the hom functor on the nose, but it lends something isomorphic. Since category theory is all upto isomorphism, shouldn't this suffice? Or are there draw backs to taking this as the definition of a representable functor? The proof sketch is roughly as follows: For any $F$ and any $x\in Fc$ , let $\Psi_x$ denote the natural transformation $R\Rightarrow F$ given by the hypothesis. Now observe that $Hom(R,R)\cong Rc$ lends $r\in Rc$ , determined by $1_R:R\Rightarrow R$ . Lemma: for any $F$ and any $x\in Fc$ , $(\Psi_x)_c(r)=x$ , and $\Psi_x$ is the unique such $\Psi$ . The naturality of $F$ implies the following diagram commutes: Chasing $1_R$ through lends the desired equality. Uniqueness follows from the isomorphism. Now by the Yoneda lemma, $r$ determines $\Phi_r : C(-,c)\Rightarrow R$ . Further, the assumption quickly lend that $1_c : c \to c$ in $C(c,c)$ determines $\Psi_ :R\Rightarrow C(-,c)$ . Now, observe that $(\Phi_r\cdot\Psi_)_c(r)=(\Phi_r)_c(1_c)=r$ , applying the lemma and then Yoneda. But, by the uniqueness claim of the lemma, we have $(\Phi_r\cdot\Psi_)=1_R$ . In the other direction, $(\Psi_\cdot\Phi_r)_c(1_c)=(\Psi_)_c(r)=1_c$ . By Yoneda and then the lemma. By Yoneda, it follows that $(\Psi_\cdot\Phi_r)=1_$ . Thus, $\Psi_:R\cong C(-,c)$ . QED Maybe this is all just a trivial consequence of the Yoneda lemma, but I have never seen hom functors, nor representable functors presented this way.

• category-theory
• representable-functor
• hom-functor

$\begingroup$ You seem on the track to rediscover the fact that a functor $R: C \to Set$ is representable if and only if its category of elements has a teminal/initial object (depending if $R$ is co or contravariant). $\endgroup$

$\begingroup$ oh, yea, I guess this is essentially saying the same thing, just different phrasing. Well, thank you! $\endgroup$

$\begingroup$ I guess I've just never seen it explicitly stated that the Yoneda lemma expresses a universal property of hom functors. $\endgroup$

$\begingroup$ I'm supposing you followed the guide I had written here. I've updated now in case you want to remove the dotted lines $\endgroup$

$\begingroup$ If you want a concise reference for the relationship among representability, Yoneda's lemma and universal properties, chapter 2 of Riehl's Category Theory in Context may be very suitable. $\endgroup$