Kernels of induced maps in cohomology

Question

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ with

$$\psi=F\circ \varphi\qquad\varphi=\psi\circ G$$

I'd like to show that under these circumstances, the induced maps in cohomology $\varphi^*,\psi^*:H^*(B,k)\to H^*(A,k)$ have equal kernels. Suppose $\alpha\in\ker\psi^*$ so that

$$0=G^*\circ\psi^*\circ F^*(\alpha)$$

By invertibility of $G^*$, it follows that $F^*(\alpha)\in\ker\psi^*$ so that $F^*$ stabilizes $\ker\psi^*$. Find $\beta\in\ker\psi^*$ such that $F^*(\beta)=\alpha$. Then

$$\varphi^*(\alpha)=\varphi^*\circ F^*(\beta)=\psi^*(\beta)=0$$

so that $\ker\psi^*\subset\ker\varphi^*$. Reverse the roles of $\varphi$ and $\psi$ to obtain equality.

Two questions:

1. Is this proof correct? If not, is the result still true?

2. Is there a result in category theory generalizing this fact? It seems I haven't used anything specific to the category of $k$-algebras and the contravariant functor $H^*(-,k)$.