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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.

An object of the slice category $\mathcal{C}/A$ is a morphism $x:X\to A$; a map from $x$ to $y:Y\to A$ is $u:X\to Y$ with $yu=x$. The identity morphism $1_A:A\to A$ is a terminal object of this slice category, even when $\mathcal{C}$ itself has no terminal object.

For a cospan $X\stackrel{u}{\to }Z\stackrel{v}{\leftarrow }Y$ in the slice category, form its underlying pullback in a category $P$ in $\mathcal{C}$, with maps $p:P\to X$ and $q:P\to Y$. If $x,y,z$ are the structure maps to $A$, then $xp=zup=zvq=yq$. This common map equips $P$ with its slice structure. An underlying mediating map into $P$ is automatically over $A$, because its composite with $xp$ is the prescribed structure map. Hence the same square is a pullback in a category in $\mathcal{C}/A$.

Part (b) applies to the slice's terminal object and pullbacks in a category. Every slice category has all finite limits when the ambient category has pullbacks in a category. The finite completeness of slice categories result needs no ambient terminal-object hypothesis.