ag.algebraic geometry – Cycle class map for singular varieties

I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of Totaro, which also does not define the map but refers to the article “Non-archimedean Arakelov theory” by Bloch, Gillet and Soule, which is published in JAG in 1995 and is not available online. Can someone give a bit more details on this map (or an alternate reference)? For example, the authors say that this map is functorial. I do not quite understand what this means, as the cohomology group is functorial under pull-back and the Chow-group is functorial under proper pushforward and flat pull-back? Moreover, does the image of the composition
$$mathrm{A}^p(X) xrightarrow{cl} mathrm{Gr}^{W}_{2p}H^{2p}(X,mathbb{Q}) xrightarrow{rest.} mathrm{Gr}^{W}_{2p}H^{2p}(X_{mathrm{smooth}},mathbb{Q})$$
is the same the usual cycle class map
$$A^p(X_{mathrm{smooth}}) xrightarrow{cl} H_{2n-2p}^{mathrm{BM}}(X_{mathrm{smooth}}) xrightarrow{P.D.}
H^{2p}(X_{mathrm{smooth}})?$$

What is the pull-back of the image of the singular cycle class map to the cohomology on
the resolution of singularities?