%
\label{prop:hom-bipolar}
For any $A, B \in \mathrm{Obj}(\tau)$,
the Hom object $[A,B]$ decomposes as
\[
    \boxed{%
    [A,B]
    \;=\;
    e_+ \cdot [A,B]_+
    \;+\;
    e_- \cdot [A,B]_-,}
\]
where:
\begin{enumerate}
    \item[\textup{(i)}]
          $[A,B]_+ := e_+ \cdot [A,B]$
          is the \textbf{B-channel component}---the
          space of B-channel projections
          of morphisms from $A$ to $B$.

    \item[\textup{(ii)}]
          $[A,B]_- := e_- \cdot [A,B]$
          is the \textbf{C-channel component}---the
          space of C-channel projections.

    \item[\textup{(iii)}]
          The two components are \textbf{independent}:
          modifying $[A,B]_+$
          does not affect $[A,B]_-$,
          and vice versa.

    \item[\textup{(iv)}]
          The decomposition is
          \textbf{functorial in both arguments}:
          for any $\tau$-morphism
          $h : B \to B'$,
          the postcomposition map
          $h_* : [A,B] \to [A,B']$
          respects the bipolar decomposition,
          i.e., $h_*([A,B]_\pm) \subseteq [A,B']_\pm$.
          Similarly for precomposition.
\end{enumerate}

By Proposition~\ref{prop:ch42-hom-tau-objects},
$[A,B]$ is a $\tau$-object.
The Idempotent Decomposition Lemma
(Lemma~\ref{lem:idempotent-decomposition}, II.L07,
Chapter~\ref{ch:idempotent-decomposition})
states that every $\tau$-object decomposes
canonically as $X = e_+ X_+ + e_- X_-$,
where $X_\pm = e_\pm X$.
Applied to $X = [A,B]$,
this gives the decomposition (i)--(ii).

\emph{Independence (iii).}
The idempotents satisfy
$e_+ \cdot e_- = 0$
(I.D21, Book~I).
Therefore
$e_+ \cdot [A,B]_- = e_+ e_- \cdot [A,B] = 0$,
and symmetrically
$e_- \cdot [A,B]_+ = 0$.
The two components live in orthogonal sub-modules of $H_\tau$,
so they are algebraically independent.

\emph{Functoriality (iv).}
Let $h : B \to B'$ be a $\tau$-morphism.
The postcomposition map
$h_* : [A,B] \to [A,B']$
is defined by $h_*(f) := h \circ f$.
Since $h$ is $\tau$-holomorphic,
$h$ commutes with the idempotent projection
(II.P09, Chapter~\ref{ch:idempotent-decomposition}):
\[
    e_\pm \cdot (h \circ f)
    \;=\;
    (e_\pm h) \circ (e_\pm f)
    \;=\;
    h_\pm \circ f_\pm.
\]
Here we use the fact that
the idempotents $e_\pm$ are central in $H_\tau$
and that composition in $\tau$ is $H_\tau$-bilinear
(it respects the $H_\tau$ module structure
at each stage).
Hence $h_*(f)_\pm = h_\pm \circ f_\pm$,
which means $h_*([A,B]_\pm) \subseteq [A,B']_\pm$.
The precomposition argument is symmetric.