Universal arrow from an object to a functor

A universal arrow from $B$ to $F:\mathcal{C}\to \mathcal{D}$ is an initial object $(A,{\eta }_B:B\to F(A))$ of $(B\downarrow F)$. The functor $F$ has a left adjoint exactly when such a universal arrow exists for every $B$; the universal property makes the chosen objects functorial.

For $F:\mathcal{C}\to \mathbf{Set}$, the comma category $(1\downarrow F)$ is the category of elements: an object is $(A,a)$ with $a\in F(A)$. Its initial objects are exactly the representations of $F$.