Cartesian square

co-universal square, pull-back square, in a category

The diagram

$$ \begin{array}{ccl} A \prod _ {S} B &\ \mathop \rightarrow \limits ^ { {p _ A}} \ & A \\ p _ {B} \downarrow \ &{} &\downarrow \ \alpha \\ B &\ \mathop \rightarrow \limits _ \beta \ &S . \\ \end{array} $$

Here $ A \prod _ {S} B $( the notation $ A \times _ {S} B $ is also used) is the fibred product of the objects $ A $ and $ B $, which is associated with

$$ \begin{array}{l} {} \\ {} \\ B \end{array} \ \begin{array}{l} {} \\ {} \\ \mathop \rightarrow \limits _ \beta \end{array} \ \begin{array}{l} A \\ \downarrow \alpha \\ S , \end{array} $$

and $ p _ {A} $ and $ p _ {B} $ are the canonical projections. The diagram

$$ \begin{array}{r} P \\ \gamma \downarrow \\ B \end{array} \ \begin{array}{l} \mathop \rightarrow \limits ^ \delta \\ {} \\ \mathop \rightarrow \limits _ \beta \end{array} \ \begin{array}{l} A \\ \downarrow \alpha \\ S \end{array} $$

is a Cartesian square if and only if it is commutative and if for any pair of morphisms $ \mu : \ V \rightarrow A $, $ \nu : \ V \rightarrow B $ such that $ \alpha \mu = \beta \nu $ there exists a unique morphism $ \lambda : \ V \rightarrow P $ which satisfies the conditions $ \mu = \delta \lambda $, $ \nu = \gamma \lambda $.

References

[1]	I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)