Product of two quotient map need not be a quotient map

Question

We know that the product of two quotient map need NOT be a quotient map. Is it any categorical explanation of this counter-intuitive phenomenon?

Answer 1

The underlying phenomenon you're observing is that the composition of a right adjoint (e.g. a product) and a left adjoint (e.g. a quotient map) does not preserve the adjunction. Perhaps this is counter-intuitive to you because left adjoints compose with each other and similarly for right adjoints, but they do not interact nicely in general.

Specifically, a quotient map (depending somewhat on the sense in which you mean the word "quotient" -- but it always refers to some sort of colimit) is a pushout of a map $f:A\to B$ along the terminal map $A\to *$, so maps out of the quotient object $B/f$ correspond to maps out of $B$ such that the subobject indicated by $f$ is sent to a point. But products have a different (dual) universal property. A map $C\to X\times Y$ to a product is the same thing as a pair of maps $C\to X$, $C\to Y$.

So the universal property of $(B/f)\times(B'/f')$ has to do with maps into the individual objects $B/f$ and $B'/f'$ (about which we can say nothing in general, since these quotient objects are defined by maps out of, rather than into, them), while $(B\times B')/(f\times f')$ has a universal property defined in terms of maps out of $B\times B'$, which we don't necessarily know anything about, since the universal property of the product doesn't tell us anything about that. These two objects have fundamentally different universal properties.