I'm working with a more or less standard definition of the category **Aut(C)** of automata over a category **C** (where **C** has finite products) which has tuples $$
A=\langle I_{A},O_{A},S_{A},\sigma_{A}, \omega_{A},q_{A}^{0}\rangle$$ as objects, where $I_{A}$,$O_{A}$ and $S_{A}$ are objects in **C**, and $\sigma_{A}:I_{A}\times S_{A}\longrightarrow S_{A}$ (the next state function), $\omega_{A}:I_{A}\times S_{A}\longrightarrow O_{A}$ (the output function) and $q_{A}^{0}:1\longrightarrow S_{A}$ (initial state) are morphisms in **C**. An automata homomorphism or *simulation* $f:A\longrightarrow B$ is a tuple $f=\langle f_{I},f_{S},f_{O}\rangle$ such that $f_{I}: I_{A}\longrightarrow I_{B}$, $f_{S}:S_{A}\longrightarrow S_{B}$ and $f_{O}:O_{A}\longrightarrow O_{A}$ are arrows in **C** and the following equalities are true: $$f_{S}\circ\sigma_{A}=\sigma_{B}\circ f_{I}\times f_{S}\tag{1}$$
$$f_{O}\circ\omega_{A}=\omega_{B}\circ f_{I}\times f_{S}\tag{2}$$
$$f_{S}\circ q_{A}^{0}=q_{B}^{0}\tag{3}$$
It's easy to proof that **Aut(C)** has finite products and it is a known fact that:

**Theorem**: If **C** is cartesian closed then **Aut(C)** is cartesian closed.

However, so far I haven't found any proof of this theorem so I'm trying to do it myself but I'm stuck on the definition of exponential objects. It seems natural to define for any pair of automata $A,B\in$ **Aut(C)** the object $B^{A}=\langle I_{B^{A}},O_{B^{A}},S_{B^{A}},\sigma_{B^{A}}, \omega_{B^{A}},q_{B^{A}}^{0}\rangle$ taking $$I_{B^{A}}=I_{B}^{I_{A}}$$
$$S_{B^{A}}=S_{B}^{S_{A}}$$
$$O_{B^{A}}=O_{B}^{O_{A}}$$
where objects on the right are exponentials in **C**, but $\sigma_{B^{A}}$ and $\omega_{B^{A}}$ are a little bit trickier because the evaluating map $\varepsilon: A\times B^{A}\longrightarrow B$ and curry $\lambda g: Z\longrightarrow B^{A}$ (for $g:A\times Z\longrightarrow B$) have to fulfill $(1)$,$(2)$ and $(3)$ in order to be morphisms in **Aut(C)**.

Any ideas or references?

PD The theorem is supposed to be proven in the following papper but I can't find it anywhere:

Cazanescu, V. 1967. "On the Category of Abstract Sequential Automata" (paper in
Romanian followed by summary in French and Russian), *Ann. Univ. Bucharest, Math. and Mechanics Series*, 16, No. 1, 31-37.

Thanks.