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Here is a way to come up with the exponential objects. Assuming that $C$ is nice enough, the forgetful functors $I,O,S : \mathrm{Aut}(C) \to C$ should have left adjoints $I^*,O^*,S^*$. For example, for every object $X \in C$, then $I^*(X)$ will be the free automaton generated by an input object $X$. Explicitly, this is given by input object $X$, state ...

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