I'm studying dg-categories, and mostly following Bernhard Keller (https://arxiv.org/abs/math/0601185). I'm trying to understand how for a pretriangulated dg-category $\mathcal{A}$, the category $H^0(\mathcal{A})$ is triangulated. The step I'm struggling with, is that first and foremost the category $H^0(\mathcal{A})$ needs to be additive, and I can't seem to find any discussion actually stating why this is the case (existence of biproducts is the problem). I have tried to come up with a proof trying to use the existence of cones granted by the property of being pre-triangulated, but there has to be something missing in my proof, as there is a crucial detail that I'm not using.
To outline the basics of the proof, I try to construct a biproduct in $Z^0(\mathcal{A})$, which then would induce the one needed in $H^0(\mathcal{A})$. To construct the biproduct of $X$ and $Y$ in $Z^0(\mathcal{A})$, I look at the morphism $0 \in Z^0(\mathcal{A})(Y[-1],X)$. The cone of this morphism (which we denote by $B$) is characterized by natural isomorphisms $\phi_Z: \mathcal{A}(Z, B) \rightarrow \mathcal{A}(Z,X) \oplus \mathcal{A}(Z,Y)$ for all $Z \in \mathcal{A}$. Using these isomorphisms I define the four morphisms
\begin{align*} i_X &:= \phi_X^{-1}(1_X, 0) \in \mathcal{A}(X,B)\\ i_Y &:= \phi_Y^{-1}(0, 1_Y) \in \mathcal{A}(Y,B)\\ (\pi_X, \pi_Y) &:= \phi_B(1_B) \in \mathcal{A}(B,X) \oplus \mathcal{A}(B,Y) \end{align*}
Then doing some calculations I confirm that these morphisms indeed satisfy the relations needed for $B$ to be a biproduct of $X$ and $Y$. However in these calculations I never use the fact that we specifically choose the cone of the zero morphism (this ensures that the differential on $\mathcal{A}(Z,X) \oplus \mathcal{A}(Z,Y)$ is a diagonal matrix, so it should be important). This means my "proof" gives an object $B_f$ for every morphism $f \in Z^0(\mathcal{A})(Y[-1],X)$, all of which satisfy the properties required to be a biproduct of $X$ and $Y$, which can't be correct.
Is my proof idea along the correct lines? If so, where should the fact that we use the zero morphism to construct the cone come in? Otherwise, are there any places where I can read about the existence of biproducts in these categories, or simpler ways to see their existence?