Representability criterion with universal element

For a categroy $\mathcal{A}$ we say that a functor $X: \mathcal{A} \to \mathbf{Set}$ is representable if there is some $A\in\mathcal{A}$ and a natural isomorphism $\alpha: \hom(A,-)\to X$.

Now as a corollary of Yoneda's Lemma we have that those representations correspond uniquely to pairs $(A, u)$ where $u\in X(A)$ and such that $u$ satisfies the condition that for each $B\in\mathcal{A}$ and $x\in X(B)$ there is a unique map $\bar x: X(A)\to X(B)$ such that $(X\bar x)(u)=x$.

I think that this condition is equivalent to the requirement that $u$ regarded as a map $\{*\}\to X(A)$ is an initial object in $\{*\}\downarrow X$.

Probably this is wrong as in my opinion this requirement is much more pregnant and it would fit well to some condition that we had for adjoints.

• You are correct: a universal element for $X$ is precisely an initial object in $(1 \downarrow X)$. – Zhen Lin Apr 14 '15 at 17:52
• Thanks! I'm a bit confused why Leinster didn't point to this connection (that's why I have asked it). – Sebastian Bechtel Apr 14 '15 at 18:34
• ps: how can I mark this thread as resolved "without answer"? – Sebastian Bechtel Apr 14 '15 at 18:35

A universal element for a functor $X : \mathcal{A} \to \mathbf{Set}$ is precisely an initial object in the comma category $(1 \downarrow X)$.
The connection with adjunctions is this: a functor $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint if and only if, for each object $C$ in $\mathcal{C}$, the functor $\mathcal{C} (C, G -) : \mathcal{D} \to \mathbf{Set}$ is representable. I leave it to you to check that $(1 \downarrow \mathcal{C} (C, G -))$ is isomorphic to $(C \downarrow G)$.
• Yes, that was wrong, there was the dual corollary for presheafs, that's probably why I mixed it, but my posts concerns functors $\mathcal{A}\to \mathbf{Set}$, right. I will correct it! – Sebastian Bechtel Apr 14 '15 at 19:20