Im working on the category of groups. Let $D$ be the push-out of the diagram $B\leftarrow A \rightarrow C$. Let $D'$ be the push-out of the diagram $B' \leftarrow A' \rightarrow C'$. It is possible to compute the push-out of the diagram $B\times B' \leftarrow A\times A' \rightarrow C\times C'$ induced by the above diagrams?
I thought that the answer was $D\times D'$ but i can't get a simple proof. Im interested in the case where $B'=\mathbb{Z}^{r_2}, A'=\mathbb{Z}^{r_1} , C'=\mathbb{Z}^{r_3}$ And the homomorphisms of $B' \leftarrow A' \rightarrow C'$ are projections.