Induced map to a product

Question

How is the map $$X_i\to Y_i \times_{M_i Y}M_iX$$ in the snippet below induced ?

Answer 1

$M_i$ is a functor together with a natural transformation $X_i\to M_iX$.

By definition of natural transformation, for any $\tau : X\to Y$, you get a commutative square :

$\require{AMScd}\begin{CD} X_i @>>> M_iX \\ @VVV @VVV \\ Y_i @>>> M_iY \end{CD}$

which, by the universal property of the pullback, induces $X_i\to Y_i\times_{M_iY} M_iX$