= Solution
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 $X\xrightarrow{u}Z\xleftarrow{v}Y$ 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>. \b[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.
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