Suppose that $\mathbf{C}$ be cartesian closed and $B$ is an object of it. We define two functors $\mathbf{C} \times \mathbf{C} \to \mathbf{C}$ by $$ C^B \times B^A \qquad\text{and}\qquad C^A$$Now I want to show that there is a natural transformation between them, i.e. with components given by $$\eta_{(A,C)} : (C^B \times B^A) \to C^A $$but how can I define such mapping? I'm a bit confused here.
Can't I just directly define $\eta_{(A,C)} (C^B \times B^A) := C^A$ and show that it is natural? Is it this trivial or I would be cheating in my definition?
Otherwise, how do you define such map? I tried manipulating evaluation and transposes, but I couldn't get anywhere.
Thanks!
By adjointness, morphisms $$C^B \times B^A \to C^A$$ are the same as morphisms $$C^B \times B^A \times A \to C$$ of which there is a canonical one, namely, the following composite: $$\require{AMScd} \begin{CD} C^B \times B^A \times A @>{\mathrm{id}_{C^B} \times \mathrm{ev}}>> C^B \times B @>{\mathrm{ev}}>> C \end{CD}$$ I leave it to you to verify naturality.