The following is taken from Sets for Mathematics by Lawvere
Definition 1: An object $S$ in a category $C$ is a $\textbf{separator}$ if and only if whenever
$${\stackrel{\small{X}\stackrel{\small\small f_1}{\stackrel{\longrightarrow}{\longrightarrow}}\small{Y}}{\small\small f_2}}{\stackrel{\stackrel{}{\stackrel{}{}}{}}{}},$$
are arrows of $C$ then
$(\forall x[S\xrightarrow{x}X\Longrightarrow f_1 x=f_2 x])\Longrightarrow [f_1=f_2]$
Definition 2: An object V is a $\textbf{cospearator}$ if for any $A$ and for any parallel pair
$${\stackrel{\small{T}\stackrel{\small\small a_0}{\stackrel{\longrightarrow}{\longrightarrow}}\small{A}}{\small\small a_1}}{\stackrel{\stackrel{}{\stackrel{}{}}{}}{}}$$
(of "generalized elements")
$(\forall \phi[A\xrightarrow{\phi}V\Longrightarrow \phi a_0=\phi a_1])\Longrightarrow [a_0=a_1]$
Notice that if $V$ is a coseparator, then $a_0\neq a_1$ entails that there is $A\xrightarrow{\phi}V$ with $\phi a_0\neq\phi a_1,$ i.e. $V$ can discriminate elements. By what we have said here, neither $V=0$ nor $V=1$ can coseparate.
Exercise: Use the fact that $1$ is a separator in the category of abstract sets to show that (in that category), if $V$ can discriminate elements, then it can discriminate generalized elements.
$\textbf{Points of confusion:}$
I am not sure how to go about the exercise because I don't know the subtle differences between when it says $V$ can discriminate elements versus $V$ being a coseparator. It seems to me that if i can discriminate elements for being in a coseparator, then it can automatically discriminate generalized elements?
From definition 1, we have $(\forall x[1\xrightarrow{x}X\Longrightarrow f_1 x=f_2 x])\Longrightarrow [f_1=f_2]$ and $V$ can discriminate elements, that means V is being in a coseparator, then again, by definition 1, letting $A=1$, $X=V$ and $\phi=x$, and also from definition 2 with $a_0=f_1$ and $a_1=f_2$, then $f_1\neq f_2$ there is $1\xrightarrow{\phi}V$ implying $\phi f_1\neq\phi f_2.$
Thank you in advance