Monomorphisms, unclear basic property, Functor

Question

Suppose that for morphisms in a category it holds that $f\circ u=v\circ f'$ and $g\circ u=v\circ g'$ and that for a functor $F$, $Fv$ is a monomorphism. Suppose that $Ff'$ and $Fg'$ are distinct. WHY it follows that $Ff$ and $Fg$ are distinct?

Answer 1

What you missed is that $Ff’$ and $Fg’$ are also assumed distinct. Thus the mono property of $Fv$ implies $FfFu=FvFf’\neq FvFg’=FgFu$. Thus we must have $Ff\neq Fg$.