Spahn couniversal mapping property 1 (Rev #2, changes)

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Let$F:X\to C$ be a functor between categories. An A object$\mathrm{<span><del\; class="diffmod">\; x</del><ins\; class="diffmod">\; C</ins></span>}\in X$ x\in C X is morphism called to satisfy auniversal mapping property with respect to $F$ and a morphism $f:a\to F(x)$$f:a\to F(x)$ if is for called any to satisfy the$y\in X$couniversal mapping property with respect to $F$, if $f=0\in C/F$ is initial in the comma category $C/F$. and any morphism $\phi:a\to G(y)$ there is a morphism $\psi:x\to y$ such that