This question is aimed at a better understanding of GIT's "categorical quotients", which are defined as coequalizers of group actions $G\times X\rightrightarrows X$ in the category of schemes. See also Anton's ~~currently unanswered~~ question about surjectivity of coequalizers, also answered by Laurent Moret-Bailly.

Suppose $f,g:W\rightrightarrows X$ and $h:X\to Y$ are scheme maps such that $hf=hg$. Let $Y_i$ be a Zariksi cover of $Y$, and let $X_i$ and $W_i$ be their pullbacks to $Y_i$ (i.e. the preimages of the open sets $Y_i$).

(a) local to global:Is it true that if $W_i\rightrightarrows X_i\to Y_i$ is a coequalizer in the category of schemes for every $i$, then $Y$ is a coequalizer in schemes?

(b) global to local:How about the converse?

**Summary** of answer by Laurent Moret-Bailly:

(a) local to global: answer is **no, but yes** if the maps on intersections $h_{ij}:X_{ij}\to Y_{ij}$ are **epic** (for example if $h$ is schematically surjective, or just universally epic).

(b) global to local: answer is simply **no**.

**Remarks**

1) The analogous statements (a) and (b) for coequalizers in the category of locally ringed spaces are **true**, which can be seen from the construction of coequalizers in LRS (coequalize the topological spaces, and take rings of invariants).

2) The analogous statements for coequalizers in the category of affine schemes is **true**: That $C\to B\rightrightarrows A$ is an equalizer is equivalent to the exactness of the $C$-module sequence $0\to C \to B \stackrel{f-g}{\to} A \to 0$, which can be checked in the localizations at prime (or maximal) ideals of $C$.

3) The analogous statements for good geometric quotients of schemes is **true**. That is, working in Schemes/$S$, if we take $W=G\times_S X$, then $X\to Y$ is a good geometric quotient iff $Y_i$ is a good geometric quotient of $W_i\rightrightarrows X_i$ for all $i$.

4) The analogous statements for *equalizers* of schemes is **true**, because fibred products can be checked/constructed on open covers, as is essentially proved in Hartshorne chapter II.3. In fact in any category, pulling back along a morphism preserves all limits, but not colimits, and in particular not coequalizers.

5) If $W=Spec(A),X=Spec(B)$ and $Y$ is their scheme coequalizer, then $Y$ is usually not affine (e.g. when gluing along opens), but $Spec(\cal{O}_Y(Y))$ is the coequalizer in the category of affine schemes. That is, $\cal{O}_Y(Y)$ is canonically isomorphic to the equalizer $C$ of $f^\sharp, g^\sharp:B\rightrightarrows A$ in rings, whose underlying set is the equalizer in sets.

6) If in (5) $B$ is a local ring, then $Y$ is affine, $Y=Spec(C)$, $C$ is local, and $C\to A$ is a local map.