Let $V_1,V_2$ be vector spaces over a same field and $W_1,W_2$ be respectively their subspaces.
Let $\mathcal V: =\{T: V_1\to V_2 \mid T(W_1)\subseteq W_2\}$.
Then there is a map $\phi: \mathcal V \to \mathcal L (V_1/W_1, V_2/W_2)$ sending $T\in \mathcal V$ to the map $\phi(T)=\tilde T: V_1/W_1 \to V_2/W_2$ defined as $\tilde T(v+W_1)=T(v)+W_2$.
My question is: is this map $\phi: \mathcal V \to \mathcal L (V_1/W_1, V_2/W_2)$ surjective ?
NOTE: For vector spaces $V,W$ by $\mathcal L(V,W)$ we mean the space of linear maps $V\to W$.
In case it helps, $\phi$ is obviously linear and I've calculated that $\ker \phi =\bigl\{T\in \mathcal V \subseteq \mathcal L(V_1,V_2) \mid T(V_1)\subseteq W_2\bigr\}$