Let h\colon A\to A' be a ring homomorphism between A,A' which are commutative rings with $1$.
Let $P,Q$ be $A$-modules. Then, are there any gaps in my following argument?
A'\otimes_{A}(P\otimes_{A}Q)=(A'\otimes_{A}P)\otimes_{A}Q=((A'\otimes_{A}P)\otimes_{A'}A')\otimes_{A}Q=(A'\otimes_{A}P)\otimes_{A'}(A'\otimes_{A}Q).
I'm concerned about this because of the proving the ring homomorphism between commutative ring with 1 induces the ring homomorphism between $K_0$ of rings.