  For a closed embedding of smooth schemes $X\hookrightarrow S$ with a fixed
first order splitting, one can construct HKR isomorphisms between the derived
scheme $X\times^R_S X$ and the total space of the shifted normal bundle
$\mathbb{N}_{X/S}[-1]$, due to Arinkin-C\u{a}ld\u{a}raru,
Arinkin-C\u{a}ld\u{a}raru-Hablicsek, and Grivaux. In this paper, we study
functoriality property of the HKR isomorphisms for a sequence of closed
embeddings $X\hookrightarrow Y\hookrightarrow S$. The HKR isomorphism is
functorial when a certain cohomology class, which we call the Bass-Quillen
class, vanishes. We obtain Lie theoretic interpretations for the HKR
isomorphisms and for the Bass-Quillen class as well.
