What's a good motivating example for the concept of a slice category? What nice example can one give a beginner to really motivate the idea of a slice category, before they've met the more general notion of a comma category? 
There's the toy example of a poset category with the slices as principal ideals -- but that doesn't exactly elicit the thought "Yep! That's pretty interesting ..."

If $X$ is a set, the slice category $\textsf{Set}/X$ can be thought of as the category of $X$-indexed collections of sets, where an object $f:Y\to X$ corresponds to the $X$-indexed collection of fibers $\{Y_x = f^{-1}\{x\}\mid x\in X\}$, and a morphism from $f:Y\to X$ to $g:Z \to X$ corresponds to a map of sets $Y_x\to Z_x$ for each $x\in X$.
This leads nicely into the observation that the fiber product $Y\times_X Z$ is just the product in the slice category, and in the category of sets it corresponds to taking the product on each fiber: $\{Y_x\times Z_x\mid x\in X\}$.
Moreover, this intuition about the meaning of the slice category is useful in other contexts, e.g. in algebraic geometry thinking of a map of schemes $X\to S$ as an algebraic family indexed by the base scheme $S$. Or in model theory (since you're a logician), thinking of a definable map of definable sets $X\to Y$ as a definable family of definable sets indexed by $Y$. Up to definable isomorphism, this can be arranged to be a projection map, so when we project the set $\phi(M)$ defined by $\phi(\bar{x},\bar{y})$ onto the $\bar{y}$ coordinates, we get the family $\{\phi(M,\bar{b})\mid \bar{b}\in \exists \bar{x}\,\phi(\bar{x},M)\}$.
