# On comparing $\mathcal L (V_1/W_1, V_2/W_2)$ with a subspace of $\mathcal L (V_1, V_2)$

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 $$\ker \phi =\bigl\{T\in \mathcal V \subseteq \mathcal L(V_1,V_2) \mid T(V_1)\subseteq W_2\bigr\}$$

We'll denote $$p_1, p_2$$ the canonical maps from $$V_1$$ (resp. $$V_2$$) onto $$V_1/W_1$$ (resp. $$V_2/W_2$$).
We can write $$V_1=W_1\oplus W'_1$$, where the subspace $$W'_1$$ is isomorphic to the quotient $$V_1/W_1$$.
Similarly, $$V_2=W_2\oplus W'_2$$, where $$W'_2\overset{\varphi}{\simeq} V_2/W_2$$.
Let $$\tau:V_1/W_1\longrightarrow V_2/W_2$$ be a linear map, and consider the section of $$p_2$$: $$V_2/W_2\xrightarrow{\enspace\varphi^{-1}\enspace} W'_2\xrightarrow{\enspace i\enspace\;}W_2\oplus W'_2=V_2$$ where $$i$$ is the canonical injection from $$W'2$$ into $$W_2\oplus W'_2$$. Set $$s=i\circ\varphi^{-1}$$. The linear map $$T=s\circ \tau\circ p_1:V_1\longrightarrow V_2$$ vanishes on $$W_1$$ and satisfies $$\: \overset{\sim}{T}=\tau$$.