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

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Let $F:X\to C$ be a functor between categories. An object $x\in X$ is called to satisfy a universal mapping property with respect to $F$ and a morphism $f:a\to F(x)$ if for any $y\in X$ and any morphism $\phi:a\to G(y)$ there is a morphism $\psi:x\to y$ such that

commutes.