The DG Yoneda embedding identifies the idempotent-completed finite twisted hull with perfect modules

A degree-$q$ module map $F:Y_X\to N$ determines $n=F_X(1_X)\in N(X{)}^q$. Conversely define $F_Z^n(a)=na$. Associativity gives $F_Z^n(ab)=(na)b$, which is precisely the defined module compatibility. The differential calculation is $(d{F}^n)(a)=d(na)-(-1{)}^{q}nda=(dn)a=F^{dn}(a)$. Identities are closed, so evaluation also commutes with differentials. Unitality gives $F_X^n(1_X)=n$, and right linearity gives $F_Z(a)=F_X(1_X)a$. Thus evaluation and $n\mapsto F^n$ are inverse cochain maps. The Hom differential preserves right linearity: the two terms involving $da$ cancel by the same Leibniz calculation.

To justify actual retract objects, let $C$ be a DG module and let $e\in H^0\mathrm{End}(C)$ satisfy $e^2=e$. Choose a closed representative $E$. Put $V={⨁}_{j\ge 0}C$ and let $S:V\to V$ send the $j$-th summand to the $(j+1)$-st by $E$. Let $T=\mathrm{Cone}(1-S)$. For any module $N$, the defining Hom complex out of $V$ is the product of those out of $C$. By AC, products commute with cohomology: choose a closed representative in each coordinate for surjectivity and a primitive in each coordinate for injectivity. The long exact sequence of the Hom complex of this cone therefore identifies its adjacent maps, in each degree, with $L:(a_j)\mapsto (a_j-a_{j+1}e)$ on ${\prod }_{j\ge 0}{\mathrm{Hom}}_H(C,N[n])$, where $H$ denotes the module homotopy category. This exact sequence follows by projecting the two-block cone Hom complex onto one block and taking kernels and boundaries.

Taking $N=Y_Y$ gives ${\mathrm{Hom}}_{\mathrm{M}\mathrm{o}\mathrm{d}-\mathcal{A}}(Y_X,Y_Y)\cong Y_Y(X)={\mathrm{Hom}}_{\mathcal{A}}(X,Y)$ as cochain complexes, so Yoneda is fully faithful before passing to cohomology.

The map $L$ is onto: for prescribed $(b_j)$ set $a_j=b_j(1-e)-{\sum }_{h<j}{b}_{h}e$. Then $a_j-a_{j+1}e=b_j$. Its kernel consists of the constant sequences $a_j=a$ with $ae=a$, since the kernel equations first imply $a_{j}e=a_j$ and then $a_j=a_{j+1}$. Exactness in step 1.2 in two consecutive degrees gives a natural isomorphism ${\mathrm{Hom}}_H(T,N)\cong \{a:C\to N:ae=a\}$. Explicitly it is restriction along $i:C\to V\to T$ to the zeroth summand. In particular $ie=i$. Taking $N=C$ gives $p:T\to C$ with $pi=e$. Since $(ip)i=ie=i$, injectivity of the same restriction with $N=T$ gives $ip=1_T$. Thus every homotopy idempotent has an actual splitting module, without requiring a strictly idempotent chain representative.

For finite direct sums and shifts, representable modules and morphism complexes are the corresponding finite direct sums and shifted complexes, so the same comparison is blockwise the isomorphism in step 2.1. For a cone $T=\mathrm{Cone}(f:X\to Y)$, its morphism complex into or out of another finite twisted object is the two-block cone complex with differential given by the same matrix $f$; applying $Y_{(-)}$ gives that identical block complex of modules. Induction on the finite number of cone steps therefore preserves the cochain-level comparison for every pair of objects in the finite twisted hull.

For two finite twisted objects, step 3.1 identifies the entire cohomology morphism spaces and preserves composition. For idempotents $e$ on $T$ and $f$ on $U$, it therefore identifies the corners $f{H}^0\mathrm{Hom}(T,U)e$. Step 2.2 supplies actual splitting modules for their images; morphisms between these splitting modules are exactly these corners, by composing with their inclusion and projection maps. This gives the claimed fully faithful extension from the idempotent completion. Its image consists precisely of retracts of finite cones, shifts and sums of representables, by the definition of perfect modules and the finite-hull comparison. No strictification of an $A_{\mathrm{\infty }}$-category is used.