Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \mathbf{Z}^2$ (on a basis $p_0,p_1$), and such that $[1](X)_{k+1}=X_k$ for all $k$. The augmentation map $\varepsilon_{[1](X)}:[1](X)_0\to \mathbf{Z}$ sends the basis elements $p_0,p_1$ to $1\in \mathbf{Z}$. For the boundary map $\partial:[1](X)_1 \to [1](X)_0$, we let $\partial(a)=\varepsilon_X(a)(p_1-p_0)$. For all other boundary maps, we let $\partial:[1](X)_{k+2} \to [1](X)_{k+1}$ be the map $\partial:X_{k+1} \to X_k$.
This functor $[1](-)$, the two-point suspension, gives an initial-object preserving functor $P:\mathbf{AugCh}\to ([1](0)\downarrow \mathbf{AugCh})$.
Then my question: Is the functor $[1](-)$ a "parametric left adjoint"? That is, is the functor $P$ a left adjoint? If this is the case, is there any explicit way to to construct the right adjoint in terms of chain complexes?
