Spahn couniversal mapping property 1 (Rev #2)

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 $F$, if $f=0\in C/F$ is initial in the comma category $C/F$.

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

commutes.