I'm trying to make sense of the following quotient: $(V/W)/(U/W)$. $V$ is a vector space over a field $\mathbb{F}$ and $W$ and $U$ are subspaces, specifically $W$ is a subspace of $U$. Naturally, the notation is suggestive that this double quotient is isomorphic to $V/U$, but I'm having some trouble proving this guess.
I construct the following square and try to make it commutative:
$\begin{array}{rcl} V&\overset{p_1}\longrightarrow&V/W\\ \\ p_2\downarrow&&\downarrow p_3\\ \\ V/U&\underset{f}\longrightarrow&(V/W)/(U/W) \end{array}$
And the bottom is my desired map, $f:V/U\rightarrow (V/W)/(U/W)$, where I define $f(v+U)=p_3\big(p_1(v)\big)=p_3(v+W)\;.$
Am I on the right track? I'm having some trouble showing that this is indeed a well-defined isomorphism...