Proposition 14

(i) Let $g_0 \in G$ and $\tau(g_0)$ be the mapping $x \mapsto f(g_0, x)$ of X into X. If $u \in \mathscr{T}^{(r)}(X)$, then $\tau(g_0)*u = \varepsilon_{g_0}*u$.

(ii) Let $x_0 \in X$ and $\varrho(x_0)$ be the mapping $g \mapsto f(g, x_0)$ of G into X. If $t \in T^{(r)}(G)$, then $\varrho(x_0)*t = t*\varepsilon_{x_0}$.

This can be proved as is Proposition 3 of no. 1.

In particular, if $u \in T(X)$ and $t \in T(G)$, $\varepsilon_{g_0} * u$ and $t * \varepsilon_{x_0}$ are equal to the products $g_0u$ and $tx_0$ defined in § 2, no. 2.