  In this paper, we investigate the property (P) that finite products commute
with arbitrary coequalizers in pointed categories. Examples of such categories
include any regular unital or (pointed) majority category with coequalizers, as
well as any pointed factor permutable category with coequalizers. We establish
a Mal'tsev term condition characterizing pointed varieties of universal
algebras satisfying (P). We then consider categories satisfying (P) locally,
i.e., those categories for which every fibre $Pt_{\mathbb{C}}(X)$ of the
fibration of points $\pi: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ satisfies
(P). Examples include any regular Mal'tsev or majority category with
coequalizers, as well as any regular Gumm category with coequalizers. Varieties
satisfying (P) locally are also characterized by a Mal'tsev term condition,
which turns out to be equivalent to a variant of Gumm's shifting lemma.
Furthermore, we show that the varieties satisfying (P) locally are precisely
the varieties with normal local projections in the sense of Z. Janelidze.
