nLab Sandbox3

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For any category $C$, there exists a covariant functor $Aut : Core(C) \to Grp$ from the core to Grp, which maps each object $A$ to an automorphism group $Aut(A)$ and each morphism $f$ to a conjugate as follows.

$$\begin{array}{rl} \mathrm{Aut}(f) : \mathrm{Aut}(A) & \longrightarrow \mathrm{Aut}(B) \\ a & \mapsto f \circ a \circ f^{-1} \\ \end{array}$$

The contravariant form can be obtained by changing the order of the conjudate as $f^{-1} \circ a \circ f$.