In Weibel's *An introduction to homological algebra* he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C_i, d)$ he defines $Cone(C)=\left(C_{i-1} \oplus C_i, \begin{bmatrix}
-d & 0 \\
-id & d
\end{bmatrix} \right)$.

It is good and it is trivial that we have a monomorphism $C\rightarrow Cone(C)$ which gives us a short exact sequance $0\rightarrow C\rightarrow Cone(C)\rightarrow C[-1]\rightarrow 0$, however we can define cone as an universal object as follows below.

For a chain complex $C$ let $F$ be a functor from chain complexes to sets given by the formula $F(D)= \left\{(f,s); f:C\rightarrow D, f=ds+sd,\text{f is a morphism}, \text{s is a chain homotopy}\right\}$ then the above cone represents it because $Hom(Cone(C),D)$ is naturally bijective to $F(D)$.

Assume now that our cone is defined as an universal object. Why do we have a monomorphism from $C$ to $Cone(C)$?

I don't know almost anything about a homotopical algebra but maybe it is a proposition there?