Let $f:A \to B$ be a homomorphism of rings. One way to define algebraic $K$ theory ($K_0$ group) is to consider the Grothendieck group of isomorphism classes of all finitely generated projective modules. Then this map $f$ induces the morphism in $K_0$ groups via the following recipe (at the level of projective modules) $P \mapsto P \otimes_{A} B$.
I don't understand where the map $f$ is invloved, it seems that only its domain and target space matters which indicates that something is wrong.
I would also like to know how this picture corresponds to the idempotent picture. Let me quickly describe what do I mean: if $P$ is projective finitely generated $A$ module then $P$ is of the form $P=eA^n$ for some idempotent $e$ and $n \in \mathbb{N}$. Then the class of $e$ viewed as an infinite matrix (completed with zeros) with respect to the equivalence class defined by similarity via some invertible matrix defines an element of $K_0$ and there is a one-to-one correspondence between this idempotent picture and the descrpition in terms of module. However in the idempotent picture map $f$ induces the map $f_*$ where $f_*([(e_{ij})_{i,j}])=[(f(e_{ij}))_{i,j}]$.
I would like to understand why these two aproaches are the same (at the level of induced maps).