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
is an equalizer. Reversing arrows gives the definition of a coequalizer. See Categories, Sections 4.10 and 4.11.
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.
Definition 35.4.2. A split equalizer is a diagram (35.4.1.1) with $g_1 \circ f = g_2 \circ f$ for which there exist auxiliary morphisms $h : B \to A$ and $i : C \to B$ such that
35.4.2.1
\begin{equation} \label{descent-equation-split-equalizer-conditions} h \circ f = 1_ A, \quad f \circ h = i \circ g_1, \quad i \circ g_2 = 1_ B. \end{equation}
The point is that the equalities among arrows force (35.4.1.1) 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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