Fix a prime $p$ and let $X$ be a $\mathbb{Z}_{p}$-scheme. Write $X_{n}:=X\otimes\mathbb{Z}/p^{n}$ and $\phi:X_{1}\rightarrow X_{1}$ for the absolute Frobenius. Let $X\hookrightarrow Z$ be a (suitable) closed immersion. Let $D_{n}$ be the PD-envelope of $X_{n}$ in $Z_{n}$ and set $J_{D_{n}}=\ker(\mathcal{O}_{D_{n}}\rightarrow\mathcal{O}_{X_{n}})$. Then for $0\leq r\leq p-1$ the syntomic complex $\mathscr{S}_{n}(r)_{X,Z}$ is the complex of étale sheaves on $X_{1}$ given by the mapping fibre of \begin{equation*} 1-\frac{\phi}{p^{r}}:\mathbb{J}^{[r]}_{n,X,Z}\rightarrow\mathbb{J}^{[0]}_{n,X,Z} \end{equation*} where $\mathbb{J}^{[r]}_{n,X,Z}$ is is the complex \begin{equation*} J_{D_{n}}^{[r]}\xrightarrow{d}J_{D_{n}}^{[r-1]}\otimes_{\mathcal{O}_{Z_{n}}}\Omega_{Z_{n}}^{1}\xrightarrow{d}\cdots \end{equation*} As Kato explains in [1], the image of $\mathscr{S}_{n}(r)_{X,Z}$ in the derived category is independent of $Z$. Kato then defines \begin{equation*} H_{\text{syn}}^{i}(X,\mathscr{S}_{n}(r)):=\mathbb{H}_{\text{ét}}^{i}(X,\mathscr{S}_{n}(r)_{X,Z}) \end{equation*} In the same paper, Kato says that this computes the syntomic cohomology defined by Fontaine-Messing using the syntomic site, but says that the details are given in [2]. But here I can only find Remark (1.1) which is the same statement, again without proof.

Does anybody have a reference that explains how these two cohomologies are the same? Or perhaps somebody can sketch how this works?

[1] *The Explicit Reciprocity Law and the Cohomology of Fontaine-Messing*, Bull. Soc. math. France,
119, 1991, p. 397–441.

[2] *On $p$-adic Vanishing Cycles (Application of ideas of Fontaine-Messing)*, Adv. Studies in Pure Math. 10, 1987, pp207-251.

Further remark:- Niziol has many (wonderful) papers studying generalisations of Kato's mapping fibre approach. I haven't been able to find an answer in any of these articles either, but in Niziol's 2006 ICM article *$p$-adic motivic cohomology in arithmetic*, there is a remark on page 7 claiming that the Fontaine-Messing construction is "philosophically" the same as the Kato construction. This only confuses me further...