The isotropy of the holonomy groupoid is the leaf holonomy group

The map is well defined and injective. Define $\mathrm{\Theta }:\mathrm{Hol}(F)(x,x)\to {\mathrm{Diff}}_x(T)$ by $\mathrm{\Theta }(\text{class of }a):=h_a(T,T)$. If $a,a^{\mathrm{\prime }}$ are holonomy-equivalent leaf loops, then by definition their holonomy germs agree, $h_a(T,T)=h_{a^{\mathrm{\prime }}}(T,T)$, so $\mathrm{\Theta }$ is well defined; conversely if the germs agree then the loops are holonomy-equivalent, so $\mathrm{\Theta }$ is injective.

The map is a homomorphism. The isotropy product is induced by concatenation, so for classes of leaf loops $a,b$ at $x$, $\mathrm{\Theta }([b]\cdot [a])=\mathrm{\Theta }(\text{class of }a\ast b)=h_{a\ast b}(T,T)=h_b(T,T)\circ h_a(T,T)=\mathrm{\Theta }([b])\circ \mathrm{\Theta }([a]),$ using multiplicativity of holonomy germs [F3]. The identity class is that of the constant loop, whose germ is the identity germ of $T$, so $\mathrm{\Theta }$ preserves identities as well, and inverses are preserved because reversal inverts the germ.

The image is the holonomy group. Every holonomy class of a leaf loop at $x$ is represented by a leaf loop $a$, and $\mathrm{\Theta }$ of its class is $h_a(T,T)={\rho }_x([a{]}^{-1})$ by [F2]; hence the image of $\mathrm{\Theta }$ is exactly ${\rho }_x({\pi }_1(L,x))=\mathrm{Hol}(L,x)$. Since $\mathrm{\Theta }$ is an injective homomorphism onto this subgroup, it is a group isomorphism onto the holonomy group.

Triviality criterion. The isotropy group is trivial exactly when its isomorphic image $\mathrm{Hol}(L,x)={\rho }_x({\pi }_1(L,x))$ is trivial, that is, exactly when ${\rho }_x$ is the trivial homomorphism. This proves the proposition and the stated criterion.