Let $f \in H^{-1}(U)$ and $u \in H^1(U)$. I know that we write $f(u)$ as the pairing $$\langle f, u \rangle_{H^{-1}, H^1}$$.

Suppose that $v$ is the weak/distributional derivative of $u$. So $$\int_0^T u\phi' = -\int_0^T v\phi$$ holds for all $\phi \in C_c^\infty(0,T)$. What does it mean to write $\langle v, w \rangle_{H^{-1}, H^1}$ where $w \in H^1(U)$? It's more than just $v(w)$, right? What's the functional that it's associated to? I ask because I see expressions like $\langle v(t), \varphi(t) \rangle_{H^{-1}, H^1}$ for $\varphi$ a $H^1(U)$-valued smooth function with compact support in $(0,T)$ and I do not know what it means.

PS: I know this is an easy question but no one answers in math.SE..