Suppose that $X$ is a Noetherian integral scheme and let $E$ be a vector bundle on $X$ (so $E$ is a locally free sheaf of finite rank). One can define the determinant of $E$, called $\det E$, as the sheafification of the presheaf given by: $$U\mapsto \wedge^{\text{max}} E(U)$$
Of course $\det E$ is a line bundle on $X$, to be more precise $\det E$ is a graded line bundle, where the "grade" is given by the rank of $E$ modulo $2$, but we don't need it here.
Now, on the book "Geometry of algebraic curves II - Arabarello, Cornalba , Griffiths", the authors at page 347 introduce the concept of determinant and list some of its properties. One property is the following one:
$\det(\cdot)$ is functorial on the base space $X$.
Can you please explain in detail what does it mean? I suppose that involes morphisms of schemes $f:Y\to X$ and it says something about $\det(f^\ast E)$.