35.4.1 Category-theoretic preliminaries
We start by recalling a few basic notions from category theory which will simplify the exposition. In this subsection, fix an ambient category.
For two morphisms $g_1, g_2: B \to C$, recall that an equalizer of $g_1$ and $g_2$ is a morphism $f: A \to B$ which satisfies $g_1 \circ f = g_2 \circ f$ and is universal for this property. This second statement means that any commutative diagram
without the dashed arrow can be uniquely completed. We also say in this situation that the diagram
Since it involves a universal property, the property of being an equalizer is typically not stable under applying a covariant functor. Just as for monomorphisms and epimorphisms, one can get around this in some cases by exhibiting splittings.
The point is that the equalities among arrows force (126.96.36.199) to be an equalizer: the map $e$ factors uniquely through $f$ by writing $e = f \circ (h \circ e)$. Consequently, applying a covariant functor to a split equalizer gives a split equalizer; applying a contravariant functor gives a split coequalizer, whose definition is apparent.
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