While it is true that Borger writes that the category does not have equalizers, what he actually explains is that the equalizer as computed in the category of rings equipped with an endomorphism (which, underlying, is the equalizer of rings) is not necessarily a Frobenius lift anymore, which is the same caveat as in Kedlaya's Execise 2.5.9(b).
It is indeed true that the category of rings equipped with a Frobenius lift admits equalizers. In fact, it is presentable: it sits in a pullback square $$ \require{AMScd} \begin{CD} \mathrm{Ring}_{\phi} @>>> \mathrm{Fun}(B\mathbb{N},\mathrm{Ring})\\ @VVV @VVV\\ \mathrm{Ring}^{\mathrm{char}=p} @>>> \mathrm{Fun}(B\mathbb{N},\mathrm{Ring}^{\mathrm{char}=p}). \end{CD} $$ The right vertical arrow takes a ring equipped with an endomorphism and mods out $p$. It is left adjoint to the obvious inclusion. The bottom horizontal arrow takes a characteristic $p$ ring and equips it with the Frobenius. It is left adjoint to the functor that takes a pair $(B,\varphi)$ consisting of a characteristic $p$ ring $B$ and a ring endomorphism $\varphi$ to the equalizer of $\varphi$ and the Frobenius. The categories in the cospan are all presentable, so the claim follows since $\mathrm{Pr}^L$ is closed under limits in $\widehat{\mathrm{Cat}}$.