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