I developed a habit of writing concisely that an object satisfies some properties $A$ and $B$ by writing $x \in A \cap B$ where $A,B$ are the categories of objects satisfying the corresponding properties. A concrete example is for a $T_0$ topological group, writing $G \in \mathrm{Grp} \cap T_0$.
Does this notation make sense mathematically? is it in use?
Namely, I am not sure if the intersection of categories can be made precise (even assuming the categories are concrete). Can it be justified? should I refrain from using such notation?