= Solution
Name the upper objects $A,B,C$ and the lower objects $D,E,F$, so the <morphisms> are $b:A\to B$, $c:B\to C$, $a:A\to D$, $g:B\to E$, $d:C\to F$, $f:D\to E$, and $e:E\to F$. The <commutative diagram> gives $gb=fa$ and $dc=eg$.
Take $u:T\to D$ and $v:T\to C$ with $efu=dv$. The right <pullback in a category> gives a unique $w:T\to B$ with $gw=fu$ and $cw=v$. The left <pullback in a category> then gives a unique $z:T\to A$ with $az=u$ and $bz=w$. It follows that $cbz=v$, giving the required factorization through the outside rectangle. Any other factorization has the same middle <morphism> by uniqueness in the right <pullback in a category>, and therefore is the same $z$ by uniqueness in the left <pullback in a category>. \b[The outside rectangle is a pullback].
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