I shall try to stick to the notation in Katz's paper. Let $k$ be a perfect field of characteristic $p>0$. Let $S_{\infty}$ be a $p$-adically complete and separated smooth formal $W(k)$-scheme and $f:X\rightarrow S_{\infty}$ a proper smooth formal $S_{\infty}$-scheme such that de Rham cohomology modules $H^{i}_{\mathrm{dR}}(X/S_{\infty}):=\mathbb{R}^{i}f_{\ast}\Omega^{\bullet}_{X/S_{\infty}}$ are locally free. Let $\nabla$ be the Gauss-Manin connection on $H^{i}_{\mathrm{dR}}(X/S_{\infty})$. Now, let $f_{1}:X_{1}\rightarrow S_{1}$ be the reduction modulo $p$ of $f:X\rightarrow S_{\infty}$. Let $\varphi_{1}:S_{1}\rightarrow S_{1}$ be the absolute Frobenius and let $\varphi:S_{\infty}\rightarrow S_{\infty}$ be a lifting of $\varphi_{1}$. Write $X'\times_{S_{\infty}}S_{\infty}$ and $X'_{1}\times_{S_{1}}S_{1}$ for the base change along $\varphi$ and $\varphi_{1}$, respectively. Let $F:X_{1}\rightarrow X_{1}'$ be the relative Frobenius. Then by functoriality we get an induced morphism (horizontal for $\nabla$)
\begin{equation*}
F:H^{i}_{\mathrm{cris}}(X'_{1}/S_{\infty})\rightarrow H^{i}_{\mathrm{cris}}(X_{1}/S_{\infty})\,.
\end{equation*}
But
\begin{equation*}
H^{i}_{\mathrm{cris}}(X'_{1}/S_{\infty})\cong H^{i}_{\mathrm{dR}}(X'/S_{\infty})\cong\varphi^{\ast}H^{i}_{\mathrm{dR}}(X/S_{\infty})
\end{equation*}
where I have used the main property of crystalline cohomology (that it computes the de Rham cohomology of a lift), base change and locally freeness. Composing with $F$ gives me a horizontal map
\begin{equation*}
F(\varphi):\varphi^{\ast}H^{i}_{\mathrm{dR}}(X/S_{\infty})\rightarrow H^{i}_{\mathrm{dR}}(X/S_{\infty})
\end{equation*}
depending on the choice of Frobenius lift $\varphi$. If $\psi$ is another lift of $\varphi_{1}$ then the connection $\nabla$ gives an isomorphism (``parallel transport'')
\begin{equation*}
\chi(\varphi,\psi):\psi^{\ast}H^{i}_{\mathrm{dR}}(X/S_{\infty})\cong H_{\mathrm{cris}}^{i}(X'_{i}/S_{\infty})\cong \varphi^{\ast} H^{i}_{\mathrm{dR}}(X/S_{\infty})\,,
\end{equation*}
and $F(\varphi)\circ\epsilon(\varphi,\psi)=F(\psi)$. Overall we get that $(H^{i}_{\mathrm{dR}}(X/S_{\infty}),\nabla, F)$ is an $F$-crystal as in (1.3) of Katz's paper. Note that $\nabla$ satisfies Griffiths transversality with respect to the Hodge filtration, so $(H^{i}_{\mathrm{dR}}(X/S_{\infty}),\nabla, \mathrm{Fil}^{\bullet}, F)$ is a filtered $F$-crystal. If the Hodge cohomology modules $H^{i}(X,\Omega_{X/S_{\infty}}^{j})$ are locally-free then $(H^{i}_{\mathrm{dR}}(X/S_{\infty}),\nabla, \mathrm{Fil}^{\bullet}, F)$ is a Hodge $F$-crystal as in 5.0 of Katz's paper.
I don't know a single reference for all of these things. Surely most of it is in P. Berthelot, Cohomologie cristalline des sch'{e}mas de caract'{e}ristique $p>0$, Lecture Notes in Math., Vol. 407. or the textbook that Carlo Beenakker suggested in the comments: P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton University Press, Princeton, NJ, University of Tokyo Press, Tokyo, 1978. There is also Remark (2.9) in P. Berthelot, A. Ogus, $F$-isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983), no. 2, 159–199. which might help you. For the Griffiths transversality bit see section 1.4 of N. Katz, Algebraic solutions of differential equations ($p$-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118. for a more general statement.
