Spahn couniversal mapping property 1 (Rev #4, changes)

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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 $C/F$. 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 $\phi:a\to G(y)$ there is a morphism $\psi:x\to y$ such that

commutes.