I feel difficult to show the existence of a map $\gamma$ which corresponds to the composition of two maps.
$\gamma$ is the following map.
$$\gamma: B^A \times C^B \to C^A$$
And I want to show that for any two maps $f: A \to B$ and $g: B \to C$, there exists a map $\gamma$ satisfying the following equation
$$\gamma(\widetilde{f}, \widetilde{g}) = \widetilde{g \circ f}$$
where I think $\widetilde{f}$, $\widetilde{g}$, and $\widetilde{g \circ f}$ are the maps from terminal object
$$\widetilde{f}: 1 \to B^A$$ $$\widetilde{g}: 1 \to C^B$$ $$\widetilde{g \circ f}: 1 \to C^A.$$
I tried to show the existence of $\gamma$ using the existence of evaluation maps $e1$, $e2$, and $e3$ satisfying the following commutative diagrams, but can't proceed the next step.
