Why convolution on a Lie group by tensor product of distributions is the same as convolution of compactly supported functions on it?

Question

Let $G$ be a Lie group. We can define the map $\mathcal E^\prime(G\times G) \to \mathcal E^\prime(G) $ by the pull-back of $C^\infty(G)\to C^\infty(G \times G)$, $\varphi \to \bar{\varphi}$. Then by a non-trivial theorem we know that $\mathcal E^\prime(G\times G) \simeq \mathcal E^\prime(G)\otimes \mathcal E^\prime(G)$. We can define the convolution of two distributions by $\langle u*v,\varphi\rangle := \langle u\otimes v, \bar{\varphi}\rangle$. It is easy to see $ \langle \delta_{g_1}*\delta_{g_2},\varphi\rangle= \langle \delta_{g_1g_2},\varphi\rangle$ -looks like finite groups-. This definition is the same as choosing a Haar measure $\lambda$ on the Lie group and identifying each compactly supported function $f$ with $\lambda f$, which is a distribution.
My question : Why the above-mentioned definition is equal to the usual way of defining the convolution by this formula: $f*g(\gamma)=\int_G f(\gamma\beta^{-1})g(\beta)d\lambda(\beta)$? I have some feeling about what is going on by the Fubini for distributions, but I can not write it in a way that it works.