Proving a linear map is surjective

Suppose $V_1, dots, V_m$ are vector spaces. Prove that
$mathcal{L}(V_1 times dots times V_m, W)$ is isomorphic to
$mathcal{L}(V_1, W) times dots times mathcal{L}(V_m, W).$ (Note that $V_{i}$‘s can be infinite-dimensional.)

I am having trouble showing that $varphi$ defined below is surjective. For every $f in mathcal{L}(V_1 times dots times V_m, W),$ I defined $f_{i}: V_{i} to W$ by $$f_{i} (v_{i}) = f (0, dots, v_{i}, dots, 0).$$ Then, I defined $varphi: mathcal{L}(V_1 times dots times V_m, W) to mathcal{L}(V_1, W) times dots times mathcal{L}(V_m, W)$ by $$varphi (f) = (f_{1}, dots, f_{m}).$$

Now, how would I show that $varphi$ is surjective?
I know I have to show that for any $(g_{1}, cdots, g_{m}) in mathcal{L}(V_1, W) times dots times mathcal{L}(V_m, W)$, there is a corresponding $g in mathcal{L}(V_1 times dots times V_m, W)$ so that $varphi (g) = (g_{1}, dots, g_{m}).$
Can I simply define $g in mathcal{L}(V_1 times dots times V_m, W)$ by
$$g (0, dots, v_{i}, dots, 0) = g_{i} (v_{i})? $$

I am not sure where to start.