$f\colon Y\rightarrow T$, $g\colon Y\rightarrow X$, $h\colon T\rightarrow S$, $i:X\rightarrow S$ be a cartesian diagram.
Then I have the diagonal morphism $\Delta\colon X \rightarrow X\times_SX$
and I also have a morphism $\phi\colon Y\times_TY \rightarrow X\times_SX$, given in natural way by the above data.
My question: what is the fibre product of $\Delta$ and $\phi$?
Is it just $\Delta\colon Y\rightarrow Y\times_TY$, the diagonal morphism of $Y$?
And another problem I have is: one can also consider $Y\times_SY$. Does this make a difference and what would I get then as fibre product of $\Delta$ and $\phi' Y\times_SY \rightarrow X\times_SX$?