Homomorphism from automorphism group of connected groupoid to outer automorphism group at a point

Statement
Consider a connected groupoid with $$X$$ its collection of points and $$P(x,y)$$ denoting the paths between points $$x,y \in X$$. Let $$x \in X$$ be a point. Denote by $$A$$ the group of automorphisms of the groupoid $$X$$, and by $$G_x$$ the group $$P(x,x)$$ under composition. Then, there is a canonical homomorphism:

$$A \to \operatorname{Out}(G_x)$$

defined as follows:


 * Given $$g \in A$$, pick a path from $$x$$ to $$y = g.x$$. Call this path $$p$$.
 * Consider now the map sending $$h \in G_x$$ to $$p^{-1} \circ (g.h) \circ p$$. Note that this map is an automorphism of the group $$G_x$$.
 * Now, the automorphism we got depended on the choice of path from $$x$$ to $$y$$. Choosing a different path gives an automorphism that differs from the original one by an inner automorphism. Hence, we get a well-defined element of $$\operatorname{Out}(G_x)$$.