Universal arrow from an object to a functor
= 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.