 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. The outside rectangle is a pullback.