I get very frustrated when an author or speaker writes "Let $X\colon= A\sqcup B$..." to mean:
- $A$ and $B$ are disjoint sets (in whatever the appropriate universe is),
- and let $X\colon= A\cup B$.
If they just meant "form the disjoint union of $A$ and $B$" this would be fine. But I've seen speakers later use the fact that $A$ and $B$ are disjoint, which was never stated anywhere except as above. You should never hide an assumption implicitly in your notation.