Connected small groupoids and vertex groups

PDF p. 6: §2.4, first paragraph

A groupoid is a category in which every morphism has a two-sided inverse. Here all groupoids are small, meaning that their objects and morphisms each form a set, and connected, meaning that they are nonempty and there is a morphism between each ordered pair of objects. Composition is read left to right: for $a:x\to y$ and $b:y\to z$, the product $ab$ means $b\circ a$.

For an object $x$ of a groupoid $\mathcal{G}$, the vertex group ${\mathcal{G}}_x={\mathrm{End}}_{\mathcal{G}}(x)$ consists of the morphisms $x\to x$, with composition as multiplication and the identity morphism $1_x$ as identity. The vertex groups at any two objects are isomorphic; the isomorphism need not be canonical.

Set-sized groupoids, functors and composition conversion

By Set-sized groupoids, functors and composition conversion, the category identities and associative composition, together with the inverses of groupoid morphisms, make the endomorphisms at $x$ a group. The left-to-right product is the same composite expressed in the indicated order.

For objects $x,y$, connectedness supplies a morphism $a:x\to y$. The map $u\mapsto a^{-1}ua$ sends ${\mathcal{G}}_x$ to ${\mathcal{G}}_y$. Associativity and $a{a}^{-1}=1_x$ show that it preserves products, and $v\mapsto av{a}^{-1}$ is its inverse. Thus it is a group isomorphism. Its construction depends on the chosen morphism $a$.