Let $\mathcal S$ be a category with finite limits. The 2-category $\operatorname{Span}(\mathcal S)$ has the same objects as are in $\mathcal S$. For objects $X,Y$, the hom category in $\operatorname{Span}(\mathcal S)$ between $X$ and $Y$ is the category of diagrams in $\mathcal S$ of the form $X \leftarrow \bullet \rightarrow Y$, and morphisms are natural transformations of such diagrams that restrict to $\operatorname{id}_X,\operatorname{id}_Y\,\,$ at the endpoints. The 1-composition of 1-morphisms is given by the obvious pull-back diagrams: $$\{X\leftarrow A \rightarrow Y\} \circ \{Y\leftarrow B \rightarrow Z\} = \{X\leftarrow A\underset Y \times B \rightarrow Z\}$$ Thinking of $\mathcal S$ as a 2-category with only identity morphisms, the "spanishization" functor (does this functor have another name?) $\mathcal S \to \operatorname{Span}(S)$ is the identity on objects and takes $\{X \overset f \to Y\}$ to $\{X = X \overset f \to Y\}$. There is also an obvious isomorphism $\mathcal S \cong \operatorname{Hom}_{\operatorname{Span}}(1,1)$, where $1 \in \mathcal S$ is the terminal object.

I believe that the correct weakened notion of "cartesian product" in a 2-category is that $X\times Y$ is determined up to equivalence (not isomorphism) as the representing object for the 2-functor $Z \mapsto \operatorname{Hom}(Z,X) \times \operatorname{Hom}(Z,Y)$, where on the right-hand side is the usual product of categories. (Incidentally, what's a good reference for $n$-Yoneda's Lemma?) Even if $\mathcal S$ has finite limits, or even all small limits, then I'm not sure whether $\operatorname{Span}(\mathcal S)$ has finite products. But for good enough categories $\mathcal S$, I feel like $\operatorname{Span}(\mathcal S)$ should also be good.

However, I believe that the product in $\operatorname{Span}(\mathcal S)$ is not the product in $\mathcal S$, i.e. spanishization does not respect limits. Provided all my beliefs are correct:

Is there an easy description of the product in $\operatorname{Span}(\mathcal S)$ in terms of $\mathcal S$? Do any products at all exist in $\operatorname{Span}(\mathcal S)$?