A functor $F : A \to B$ is called very rich if for all $A,A\tick\in A_o$, and $b : FA\to FA\tick$, there exists a span $A\xleftarrow{r} V \xto{r\tick} A\tick$ in $ A$ such that $Fr$ is invertible and $b = Fr\tick\circ (Fr)^{-1}$.
A functor $F : A \to B$ is called very rich if for all $A,A\tick\in A_o$, and $b : FA\to FA\tick$, there exists a span $A\xleftarrow{r} V \xto{r\tick} A\tick$ in $ A$ such that $Fr$ is invertible and $b = Fr\tick\circ (Fr)^{-1}$.