Making one leg cartesian

Fix a span \(B \leftarrow A \to C\). A span over this span is a diagram of the form

Given such a span, we define two new spans over \(B \leftarrow A \to C\) as follows.

1. The span obtained by making the left leg cartesian is

   \[Q^{\ell} := Q, \qquad P^{\ell} := Q \times_B A, \qquad R^{\ell} := P^{\ell} \sqcup_P R .\]

2. The span obtained by making the right leg cartesian is

   \[R^{r} := R, \qquad P^{r} := R \times_C A, \qquad Q^{r} := P^{r} \sqcup_P Q .\]