Assuming we have two algebraic groups $G_1$ and $G_2$ over $k$. Then the direct product $G_1 \times G_2$ with the direct product group structure is an algebraic group. Is this the same as the fibre product $G_1 \times_k G_2$?
I think it should be, but fail to give a precise argument..
The product in a category which has a terminal object is the fiber product over the terminal object.