In a group $G$ , the conjugation of any element $g \in G$ by $h$ defines an inner automorphism $\psi_h : g \to hgh^{-1}$.
If one considers instead a connected groupoid $\mathcal{G}$, is there a similar definition for the conjugation of the morphisms of $\mathcal{G}$ by a particular morphism $\mu_h \in \mathcal{G}$ ?
The correct analogue is the following one. At each object $x$ of your groupoid choose an element $\mu_x\in Hom(x,x)$. Call this entire construct $\mu$. You can now define a conjugation by $\mu$ by mapping any morphism $g:x\to y$ in the groupoid to $\mu_y^{-1}\circ g \circ \mu_x$. It is easily checked that this gives rise to an automorphism of the groupoid and that it extends the notion of inner automorphism for groups when viewed as one-object groupoids.