I was wondering is there an isomorphism $$\mathrm{Hom}\big(\mathrm{cone}(A\xrightarrow{f}B),\,C\big)\,\cong\,\mathrm{cone}\big(\mathrm{Hom}(B,\,C)\xrightarrow{f^\ast}\mathrm{Hom}(A,\,C)\big)\,?$$ On the level of graded vector spaces, one has the following degree one isomorphism: $$\mathrm{Hom}(A[1]\oplus B,\,C)\,\cong\,\mathrm{Hom}(A[1],\,C)\oplus \mathrm{Hom}(B,\,C)\,\cong\,\mathrm{Hom}(A,\,C)\oplus \mathrm{Hom}(B,\,C)[1]\,.$$ I tried to check the differentials, but the signs didn't match.
If the answer of the question above is positive, are there any conceptual reasons behind that? I guess mapping cone can be viewed as "homotopical cokernel".
Any helps or comments would be very appreciated.