Let $\mathcal{G}=[G_1 \rightrightarrows G_0]$ and $\mathcal{H}=[H_1 \rightrightarrows H_0]$ be two Lie groupoids. Consider the product category $\mathcal{G} \times \mathcal{H}$.
My question:
Is the product category $\mathcal{G} \times \mathcal{H}$ always a Lie groupoid with respect to the standard structure maps induced from the structure maps of Lie groupoids $\mathcal{G}$ and $\mathcal{H}$?
Particularly, I am not able to convince myself that source map $s_{\mathcal{G}} \times s_{\mathcal{H}}: G_1 \times H_1 \rightarrow G_0 \times H_0$ and target map $t_{\mathcal{G}} \times t_{\mathcal{H}}: G_1 \times H_1 \rightarrow G_0 \times H_0$ are always submersions. (Here $s_{\mathcal{G}}$, $t_{\mathcal{G}}$ are source and target maps of the Lie groupoid $\mathcal{G}$ and $s_{\mathcal{H}}$, $t_{\mathcal{H}}$ are source and target maps of the Lie groupoid $\mathcal{H}$ )
If the answer to my question is negative, then under what conditions on $\mathcal{G}$ and $\mathcal{H}$ the product category $\mathcal{G} \times \mathcal{H}$ is a Lie groupoid with the induced structure maps?