The Clowder Project

where $-_{1}\times -_{2}$ is the functor where

• Action on Objects. For each $\webleft (A,B\webright )\in \text{Obj}\webleft (\mathsf{Sets}\times \mathsf{Sets}\webright )$, we have
  \[ \webleft [-_{1}\times -_{2}\webright ]\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A\times B. \]

• Action on Morphisms. For each $\webleft (A,B\webright ),\webleft (X,Y\webright )\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on $\textup{Hom}$-sets
  \[ \times _{\webleft (A,B\webright ),\webleft (X,Y\webright )} \colon \mathsf{Sets}\webleft (A,X\webright )\times \mathsf{Sets}\webleft (B,Y\webright )\to \mathsf{Sets}\webleft (A\times B,X\times Y\webright ) \]

  of $\times $ at $\webleft (\webleft (A,B\webright ),\webleft (X,Y\webright )\webright )$ is defined by sending $\webleft (f,g\webright )$ to the function

  \[ f\times g\colon A\times B\to X\times Y \]

  defined by

  \[ \webleft [f\times g\webright ]\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f\webleft (a\webright ),g\webleft (b\webright )\webright ) \]

  for each $\webleft (a,b\webright )\in A\times B$.