I understand that the product in a slice category $\mathscr{C}/X$ is the pullback in the category $\mathscr{C}$ - my end goal is to prove that fact. The one thing that is (currently) making me bash my head is what the product in that slice category even looks like - I've always had trouble wrapping my mind around categories whose objects are arrows.
Would anyone be able to help me visualize the product in some meaningful, useful way?
This is just a normal pullback square in some category, arranged a bit differently:
$$ \begin{array}{ccccc} a & \leftarrow & a \times_c b & \rightarrow & b \\ & \searrow & \downarrow & \swarrow \\ & & c \end{array}$$
Now in the slice over $c$, each arrow to $c$ gets turned into an object. Let's call them $f$, $g$ and $h$ from left to right in the above diagram. The arrows $a \times_c b \rightarrow a$ und $a \times_c b \rightarrow b$ each make a triangle with two of the arrows $f$, $g$ and $h$ commute, so in the slice we have arrows
$$(a,f) \leftarrow (a \times_c b, g) \rightarrow (b,h)$$
which are projections making $(a \times_c b,g)$ into a candidate for the product $(a,f) \times (b,h)$ in the slice. Now the universality of $(a \times_c b,g)$ can be seen by adding in another candidate for the product in the slice and transfering that into the diagram in the original category, where it becomes a candidate for the pullback $a \times_c b$.