I'm using notation close to Street-Walters *"Yoneda structures"*.

For any locally small category $\textbf{A}$
there are, of course, $\hat{\textbf{A}}:=\textbf{set}^{\textbf{A}^{op}}$ and
$\check{\textbf{A}}:=(\textbf{set}^{\textbf{A}})^{op}$
as well as the corresponding Yoneda embeddings
$Y(\textbf{A}):\textbf{A}\rightarrow\hat{\textbf{A}}$ and
$Z(\textbf{A}):\textbf{A}\rightarrow\check{\textbf{A}}$.

For any locally small functor $F:\textbf{A}\rightarrow\textbf{B}$, there is a covariant $F$-*weighted-hom functor* $\textbf{B}(F,1):\textbf{B}\rightarrow\hat{\textbf{A}}$ as well as a contravariant one $\textbf{B}\langle1,F\rangle:\textbf{B}\rightarrow\check{\textbf{A}}$
and evaluation of $F$ on arrows can be encoded via a natural transformation from the covariant Yoneda
$\chi^F:Y(\textbf{A})\Rightarrow\textbf{B}(F,1)F$ or via one from the contravariant one $\psi^F:Z(\textbf{A})\Rightarrow\textbf{B}\langle 1,F \rangle F$.

It is a fact that, in $\textbf{Cat}$, the covariant $F$-weighted-hom is a left Kan extension along $F$ of the covariant Yoneda of its domain (SW Axiom 1 for the Yoneda structure of $\textbf{Cat}$) and that $F$ is the *absolute* left lifting of Yoneda along the $F$-weighted-hom (SW Axiom 2), namely:

$(1) \hspace{12pt} (\textbf{B}(F,1),\chi^F)=lan_FY(\textbf{A})$

$(2) \hspace{12pt} (F,\chi^F)=LIF_{\textbf{B}(F,1)} Y(\textbf{A})$

By pedantically adapting the proof of $(1)$ I can see its contravariant version:

$(1*) \hspace{8pt}(\textbf{B} \langle 1,F \rangle,\psi^F)=lan_FZ(\textbf{A})$

Is $(1*)$ a consequence of $(1)$ in the sense that there is a direct way, that I guess should pass through the underlying profunctors, to prove $(1*)$ assuming $(1)$ ?

Is there a contravariant version of $(2)$ ?