category: universal construction [[!redirects universal mapping property 1]] 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 $F$* 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 [[nLab: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 $$\array{ a&\stackrel{f}{\mapsto}&F(x) \\ &\searrow^\phi&\downarrow^{G(\psi)} \\ &&G(y) }$$ commutes.