The following question is distantly related to the Fundamental Homomorphism Theorem:
$\fbox{Let}$
$f:A \rightarrow B$
$g_1:B \rightarrow C$
$g_2:B \rightarrow C$
and finally assume $g_1 \circ f = g_2 \circ f$
$\fbox{WTS}$
$g_1 = g_2$
$\fbox{Proof}$
Let $b \in B$ and consider that there must exist an $a \in A$ s.t. $g_1(b) = (g_1 \circ f)(a)$ so that
$ g_1(b) = (g_1 \circ f) (a) = (g_2 \circ f) (a) = g_2(b) $
But I'm unclear on the very last assertion that $(g_2 \circ f) (a) = g_2(b)$. Couldn't it be that $(g_2 \circ f) (a) = g_2(b_1)$ s.t. $b_1 \ne b$? Does this mean the "proof" is wrong?