Pullback of Commutative Triangle

Theorem
Let $\mathbf C$ be a metacategory.

Suppose that the following is a commutative diagram in $\mathbf C$:


 * $\begin{xy}\xymatrix@+1em@L+2px{

A' \ar[rr]^*+{h_\alpha} \ar[dd]_*+{\alpha'} & & A \ar[rd]^*+{\gamma} \ar[dd]^(.4)*+{\alpha}

\\ & B' \ar[ld]^*+{\beta'} \ar[rr] |{\hole} ^(.3)*+{h_\beta} & & B \ar[ld]^*+{\beta}

\\ C' \ar[rr]_*+{h} & & C }\end{xy}$

and that the two squares in it are pullback diagrams.

Then there is a unique morphism $\gamma': A' \to B'$ making the following commute:


 * $\begin{xy}\xymatrix@+1em@L+2px{

A' \ar[rr]^*+{h_\alpha} \ar[dd]_*+{\alpha'} \ar@{-->}[rd]^*+{\gamma'} & & A \ar[rd]^*+{\gamma} \ar[dd]^(.4)*+{\alpha}

\\ & B' \ar[ld]^*+{\beta'} \ar[rr] |{\hole} ^(.3)*+{h_\beta} & & B \ar[ld]^*+{\beta}

\\ C' \ar[rr]_*+{h} & & C }\end{xy}$

Furthermore, $\gamma'$ makes the following a pullback:


 * $\begin{xy}\xymatrix@+.5em@L+2px{

A' \ar[r]^*+{h_\alpha} \ar[d]_*+{\gamma'} & A \ar[d]^*+{\gamma}

\\ B' \ar[r]_*+{h_\beta} & B }\end{xy}$