  For a variety $\mathcal{V}$, it has been recently shown that binary products
commute with arbitrary coequalizers locally, i.e., in every fibre of the
fibration of points $\pi: \mathrm{Pt} (\mathbb{C}) \rightarrow \mathbb{C}$, if
and only if Gumm's shifting lemma holds on pullbacks in $\mathcal{V}$. In this
paper, we establish a similar result connecting the so-called triangular lemma
in universal algebra with a certain categorical $\textit{anticommutativity}$
condition. In particular, we show that this anticommutativity and its local
version are Mal'tsev conditions, the local version being equivalent to the
triangular lemma on pullbacks. As a corollary, every locally anticommutative
variety $\mathcal{V}$ has directly decomposable congruence classes in the sense
of Duda, and the converse holds if $\mathcal{V}$ is idempotent.
