Spahn couniversal mapping property 1

Let $F:X\to C$ be a functor between categories. A $C$ morphism $f:a\to F(x)$ is called to satisfy the couniversal mapping property with respect to $F$ or we say that $f$ is a couniversal arrow from $c$ to $F$ or that $f$ is the initial arrow from $c$ to $F$, if $f=0\in C/F$ is initial in the comma category $c/F$.

This means: for any $y\in X$ and any morphism $\phi:a\to F(y)$ there is a morphism $\psi:x\to y$ such that

$$\array{ a&\stackrel{f}{\mapsto}&F(x) \\ &\searrow^\phi&\downarrow^{F(\psi)} \\ &&F(y) }$$

commutes.