Base change of classical varieties when the pullback exists

For maps $f:X\to S$ and $g:T\to S$, a classical base change in a fixed classical category is an object $X{\times }_{S}T$ with maps to $X,T$ commuting over $S$, universal among all commuting pairs in that same category. A closed equalizer locus in an already-constructed product is only a candidate until both its algebraic-set structure and this universal property have been checked in the chosen category. In particular, an affine-variety product theorem does not by itself construct a reducible equalizer in the larger algebraic-set category. A constructed base change need not be a variety: it can be reducible or empty. This is not a definition of arbitrary classical or scheme fibre products.