Slice category ($\mathcal{C}/A$)

The slice over $A$ has morphisms $x:X\to A$ as objects and commuting triangles $yu=x$ as arrows $u:(X,x)\to (Y,y)$. Its terminal object is $1_A$, regardless of whether $\mathcal{C}$ has a terminal object. It is the comma category $(1_{\mathcal{C}}\downarrow A)$, viewing $A$ as a functor from the terminal category.

A natural transformation $P\to F$ assigns a set of points over every element $x\in F(c)$. These fibers form a functor on the category of elements of $F$. Conversely, a functor on the category of elements gives $P(c)$ as the disjoint union of its values over all $x\in F(c)$. The two constructions are inverse up to natural isomorphism. Smallness of $\mathcal{C}$ makes the category of elements small and allows this description to transfer Cartesian closure to every slice.

If a category has pullbacks in a category, each slice category is finitely complete. The identity of the base object is terminal in the slice. Ambient pullbacks in a category inherit a unique common structure map to the base and satisfy the same universal property over that base. A terminal object and pullbacks in a category construct finite products in a category and equalizers, hence all finite limits.