I looked for material relating to compositions of equivalence relations, and was surprised to find the claim (here) that the composition of equivalence relations is not necessarily again an equivalence relation. Upon reading more closely, I saw that the author of that post was using the term "composition", in the context of equivalence relations, rather differently to the interpretation that seemed natural to me. That author uses "composition" to mean a certain binary operation on the set of binary relations on a given set.
My assumed notion of "composition" of equivalence relations is this: let $S$ be a set, let $\sim$ be an equivalence relation on $S$, and let $\bowtie$ be an equivalence relation on $S / \sim$. Given an $s \in S$, write $[s]_{\sim}$ for the equivalence class of $s$ under $\sim$. Now define an equivalence relation $\bowtie \circ \sim$ on $S$ as follows: given any $s \in S$, $s \bowtie \circ \sim t$ if and only if $[s]_{\sim} \bowtie [t]_{\sim}$.
The reason why this makes sense as a notion of "composition" is that the quotient map associated to the composition of equivalence relations is the composition of the quotient maps of the individual equivalence relations, after a "flattening" step in which each equivalence class is replaced by the union of its elements (those elements themselves being the equivalence classes under the first equivalence relation). Why is the term "composition" used to mean something else for equivalence relations?