Let $f\colon A\to B$ be a homotopy equivalence of chain complexes (with $g$ the weak inverse and chain homotopies $\phi\colon {\rm id}_D\to f\circ g$ and $\psi\colon g\circ f\to{\rm id}_C$). Then the associated chain map between Hom-complexes $$ f_\ast\colon\mathcal{H}om(X,A)\to\mathcal{H}om(X,B) $$ is again a homotopy equivalence with weak inverse $g_\ast$ and chain homotopies $\phi_\ast$ and $\psi_\ast$.
I think this is also true for a general dg-category $\mathcal{C}$, namely if $f\colon A\to B$ is a homotopy equivalence inducing isomorphism in the homotopy category, then the associated chain map $$ f_\ast\colon\mathcal{H}om_{\mathcal{C}}(X,A)\to\mathcal{H}om_{\mathcal{C}}(X,B) $$ is a homotopy equivalence. Now, all the above constructions $f_\ast$, $g_\ast$, $\phi_\ast$ and $\psi_\ast$ still exist. The problem is to show that $\phi_\ast$ and $\psi_\ast$ are chain homotopies.
Note that, in the case of chain complexes, for $h$ an element of $\mathcal{H}om(X,A)$, we have \begin{align} (\partial\circ\phi_\ast+\phi_\ast\circ\partial)(h) &= \partial(\phi\circ h) +\phi\circ(\partial (h)) \\ &= \partial^D\circ\phi\circ h - (-1)^{1+|h|}\phi\circ h\circ\partial^X \\ &\qquad + \phi\circ\partial^C\circ h - (-1)^{|h|}\phi\circ h\circ\partial^X \\ &= (\partial^D\circ\phi + \phi\circ\partial^C)\circ h \\ &= (g\circ f - {\rm id}_C)\circ h \\ &= (g_\ast\circ f_\ast -{\rm id})(h). \end{align} Hence $\phi_\ast$ is a chain homotopy. Similar for $\psi_\ast$.
My question is: How to show $\phi_\ast$ is a chain homotopy in general dg-category $\mathcal{C}$ ?