Let $\mathcal{C}$ be a category with pullbacks and consider its associated span (bi)category $\text{Span}(\mathcal{C})$. I'm trying to guess how products in $\text{Span}(\mathcal{C})$ look like but I'm a bit stuck. First of all, we have to make a difference between product and strict product. My guess for the product is that it is the same product as in $\mathcal{C}$ but I'm not completely sure, so I would like to verify if I'm right or wrong and also some hint about the non-strict case would be appreciated.
The problem with the strict case is that well, if one tries to show it, it is easy to write the product diagram in $\mathcal{C}$ as a diagram in $\text{Span}(\mathcal{C})$: just expand the diagram in $\mathcal{C}$ adding identities to create spans, but the problem comes when one tries to show the universal property, since I don't see how to use the universal property in $\mathcal{C}$.