In a dg-category $\mathcal{C}$, the *$n$-translation* of an object $C$ is an object $C[n]$ representing the functor
$$
{\rm Hom}(-,C)[n].
$$
The *cone* of a closed morphism $f\colon C \to D$ of degree zero is an object
${\rm Cone}(f)$ representing the functor
$$
{\rm Cofiber}\big({\rm Hom}(-,C)
\stackrel{{f_{\ast}}}{\longrightarrow}{\rm Hom}(-,D)\big).
$$
On the other hand, the *homotopy cofiber* of $f$ is an object
${\rm Cofiber}(f)$ representing the functor
$$
{\rm Fiber}\big({\rm Hom}(D,-)
\stackrel{{f^{\ast}}}{\longrightarrow}{\rm Hom}(C,-)\big).
$$

Now, suppose $\mathcal{C}$ has zero object, all translations of all objects, and all cones of all morphisms.

My question: **Is there any easy way to show ${\rm Cofiber}(f)$ and ${\rm Cone}(f)$ are isomorphic?**

To be more efficiently, I know that there are canonical closed morphism $\iota\colon D\to{\rm Cone}(f)$ of degree $0$ and morphism $h\colon C\to{\rm Cone}{f}$ of degree $-1$. They induce a natural cochain map $$ {\rm Hom}\big({\rm Cone}(f),-\big) \longrightarrow {\rm Fiber}\big({\rm Hom}(D,-) \stackrel{{f^{\ast}}}{\longrightarrow}{\rm Hom}(C,-)\big) $$ which sends any $x\colon{\rm Cone}(f)\to X$ to the pair $(x\circ\iota,x\circ h)$.

However, I don't know how to finish the proof, i.e. show this is an isomorphism.