I have an object in the slice / arrow category $\mathbf{Set}/X^2$ and I want to transport it to $\mathbf{Set}/X$ by forgetting the second element of the index. Reading up on slice categories I've only always encountered the induced base change/substitution functor, but that's not what I am doing here.
I am trying to define a head-indexed List shape/base functor, so for element type $A$, I'm slicing over $A + 1$, and the reason I need this inclusion is because I have: $L(X \to^{f} (A+1)):= (1\to^{\pi_1} (1+A)) + \text{Mystery functor here }(A\to^{\pi_2} (1+A))\times (X \to^{f} (A+1))$ (where $\times$ is the product in $\mathbf{Set}^{\to}$, so the arrow category.)
What would be the "right" categorical way to go about this?
Yes, there is such a functor. In general, let $\mathscr{C}$ be a category, $A$ and $B$ be two objects in that category. For any morphism $f\colon A\to B$ in $\mathscr{C}$, $f$ induces a functor on the slice categories $f_*\colon\mathscr{C}/A\to\mathscr{C}/B$ by sending any $g\colon L\to A$ (as an object in $\mathscr{C}/A$) to $f\circ g\colon L\to B$, which is an object in $\mathscr{C}/B$. I will let you work out how this functor acts on morphisms, and that the compatibility conditions for this to actually be a functor do hold. (Hint: the morphism obtained via pushforward will be described by the same actual function of sets; you just need to check that the new diagram commutes, which will be straightforward.)
In your case, you have the slice category over $X^2=X\times X$ (where $X^2$ is your "$A$" in the above) and the slice category over $X$ (which is your "$B$" in the above). Let $\pi_1\colon X\times X\to X$ be the function which sends a pair $(x_1,x_2)$ to the first coordinate $x_1$ (this is your "$f$"). Then $\pi_1$ induces a functor on your slice categories, which behaves as you would like.