$\require{AMScd}$ Consider a symmetric monoidal category $(\mathcal{C}, \otimes)$ with an adjunction $(A\otimes B, C) \simeq (A, C^{B})$ id est an internal hom set functor.
For every object $X$ there is a canonical arrow $\begin{CD}X^X\otimes X^X @>{m}>>X^X\end{CD}$ given as the adjunct of $\begin{CD}X^X \otimes X^X \otimes X @>{X^X \otimes \: \text{ev}}>> X^X \otimes X @>{\text{ev}}>> X \end{CD}$ where $\text{ev}$ is the evaluation map.
In $\mathbf{Set}$ this arrow gives $X^X$ a monoid structure. I'm wondering if it is true in the general case. I tried to prove it by drawing a lot of diagrams, but failed, probably from want of a global intuition.
A related question (though the first one seems more important) is to prove that $X\otimes X \overset{f}{\rightarrow} X$ defines an associative law if and only if the diagram
$\begin{CD} X\otimes X@>{f^{adj}}>> X^X \otimes X^X \\ @VV{f}V @VV{m}V \\ X @>{f^{adj}}>> X^{X} \end{CD}$
commutes. In $\mathbf{Set}$ it means that multiplication by $a$ followed by multiplication by $b$ is the same as multiplication by $ba$.
All of these questions arised when trying to prove both definitions of $\mathbb{N}$ as an universal pointed arrow or an universal pointed monoid are equivalent.