Suppose you have two objects $A$ and $A'$ in a category $\mathfrak{C}$, and morphisms $i_C:\text{Mor}(C,A)\to\text{Mor}(C,A')$ for any object $C\in\mathfrak{C}$. Show that the $i_C$ are induced from a unique morphism $g:A\to A'$. More precisely, show that there is a unique morphism $g:A\to A'$ such that for all $C\in\mathfrak{C}$, $i_C$ is $u\to g\circ u$.
That $g$ is unique seems wrong to me! Taking a simple example in $\text{Set}$ would show this. Any help would be greatly appreciated.
Edit: The exercise says "The morphisms $i_C:\text{Mor}(C,A)\to\text{Mor}(C,A')$ commute with the maps of the form $\text{Mor}(C,A)\to\text{Mor}(B,A)$." I don't know what this means.