Products, coproducts, and equalizers

A product of $A$ and $B$ consists of an object $P$ and projections to the two factors. Given any test object $T$ with maps to both factors, there must be a unique map from $T$ to $P$ producing those two maps after projection. In Set, ordered pairs provide this map point by point. The universal property characterizes the product with its projections, not an arbitrary object having the same number of elements.

For each $T$, the property is a bijection from $\mathcal C(T,P)$ to a product of Hom sets, natural in the test object. Empty products are terminal objects because the empty family of maps has exactly one compatible realization. A category need not possess products at all. When two products do exist, applying their mapping properties to each other produces inverse comparison maps, uniquely determined by their compatibility with both projections.

A coproduct has injections from $A$ and $B$ and is universal for extending a pair of maps out of these objects to one map out of the coproduct. In Set, a tagged disjoint union keeps an element of the first summand distinct from an equal-looking element of the second. The tags are needed because the two original maps may prescribe different outputs at that shared-looking element. An overlapping ordinary union would accidentally force those outputs to agree.

The mapping property now tests maps into an arbitrary target $T$. This reversal distinguishes a coproduct from a product even when a particular category happens to realize both with the same underlying object. Empty coproducts are initial objects. A map from a coproduct is determined by its restrictions to summands; a map into a product is determined by its projections. Neither statement authorizes interchanging the two directions in an unfamiliar category.

For parallel arrows $f,g:X\to Y$, an equalizer is an arrow $e:E\to X$ with $f\circ e=g\circ e$. It must satisfy more than this equation: whenever $h:T\to X$ also equalizes the pair, there is exactly one $u:T\to E$ with $e\circ u=h$. In Set, take the subset of points at which the functions agree and use its inclusion. Every equalizing function lands in that subset, so restricting its codomain supplies the factorization.

This construction is a subobject of the source, while the dual coequalizer is a quotient of the target identifying outputs that must agree. Confusing them reverses the equation being imposed. In a general category an equalizer arrow is monic: if two maps become equal after composition with it, they are two factorizations of the same equalizing map and hence coincide. The familiar injectivity of a subset inclusion is therefore an instance of a universal argument.