Base change of classical varieties when the pullback exists

Fix a category $\mathcal{C}$ of classical algebraic objects with its specified morphisms. For morphisms $f:X\to S$ and $g:T\to S$ in $\mathcal{C}$, a classical base change, when it exists, is the pullback of Pullbacks and pushouts as limits and colimits of cospans and spans in $\mathcal{C}$: an object $X{\times }_{S}T$ with morphisms to $X,T$ commuting over $S$, universal among all commuting pairs from objects of that same category. This is a conditional definition and uses no product-existence theorem. 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. In a category allowing reducible or empty algebraic sets, a base change can have either feature; in a category restricted to nonempty irreducible varieties, any pullback must itself meet that restriction and its existence must be checked there. No assertion of arbitrary classical or scheme fibre-product existence is made.