Universal property of a fibre product
= Universal property of a fibre product
For morphisms $X\to S$ and $Y\to S$, maps $T\to X\times_SY$ are naturally equivalent to pairs of maps $T\to X$ and $T\to Y$ whose composites to $S$ agree.
= Universal property of a fibre product
For morphisms $X\to S$ and $Y\to S$, maps $T\to X\times_SY$ are naturally equivalent to pairs of maps $T\to X$ and $T\to Y$ whose composites to $S$ agree.