definition. functorial in [leinster2016basic, sec. 4.1] [tt-0041]

✍️source

For some expression \(E(X)\) containing \(X\), when we say \(E(X)\) is (covariant) functorial in \(X\), we mean that there exists a functor \(\mathscr {F}\) such that for every \(f : X \to X'\).

Dually, we use the term contravariantly functorial in.