In category theory, the exponential object $X^Y$ is defined, roughly, by saying that if we take some other object $Z$ and the product between $Z$ and $X$, then any morphism from that product to $Y$ has an equivalent morphism that “goes through” $X^Y$.
However, what I find weird about this construction (note that category theory in general is weird to me), is that we need a third object $Z$ and the assumption that a cartesian product exists between them, to define the exponential object $X^Y$.
Is it not possible to define it more directly somehow? Intuitely it seems like we don’t need to talk about $Z$ or about cartesian products, to talk about functions from $X$ to $Y$ (I know that category theory is not only about functions, but this is how I intuitively think about it).