In Kedlayas lecture notes on prismatic cohomology it is an exercise (2.5.9) to prove that the category of rings equipped with a Frobenius lift (denotet $\mathbf{Ring}_\phi$, the morphisms are ringhoms compatible with the corresp. Frob. lift) admits equalizers. However in a lecture of James Borger on $\delta$-rings he claims that $\mathbf{Ring}_\phi$ doesn't have equalizers. Does someone know what is true and how to prove it or what would be a counterexample.
Link to Kedlayas work: https://kskedlaya.org/prismatic/sec_overview.html\ Link to Borgers lecture: https://www.youtube.com/watch?v=_y2Tcu-iJV4&t=10s\ Any help or advice would be great.

In Kedlayas lecture notes on prismatic cohomology it is an exercise (2.5.9) to prove that the category of rings equipped with a Frobenius lift (denotet $\mathbf{Ring}_\phi$, the morphisms are ringhoms compatible with the corresp. Frob. lift) admits equalizers. However in a lecture of James Borger on $\delta$-rings he claims that $\mathbf{Ring}_\phi$ doesn't have equalizers. Does someone know what is true and how to prove it or what would be a counterexample.
Link to Kedlayas work: https://kskedlaya.org/prismatic/sec_overview.html
Link to Borgers lecture: https://www.youtube.com/watch?v=_y2Tcu-iJV4&t=10s
Any help or advice would be great.