*I'm asking this question in the most model-ambiguous way I can since this is the kind of answer i'm looking for.*


There are various explicit constructions of the Whitehead and Postnikov towers. I'm trying to understand what exactly characterizes these construction.



> 
> **Postnikov tower:** A Postnikov tower of (a possibly non-pointed) space is a factorization of the terminal morphism $X \to \*$ to a directed limit diagram:
> 
> 
> $$X \cong\underset{\rightarrow\_n}{\lim}X\_n \to \dots \to X\_2 \to X\_1 \to X\_0 \cong \*$$
> Such that $X\_k$ is $(k-1)$-truncated and each morphism $X\_{k} \to X\_{k-1}$ is a $k$-equivalence (isomorphism on homotopy groups in degree smaller than $k$ and surjection on $k$).
> 
> 
> **Question 1:** Does this property determine the Postnikov tower up to weak equivalence of diagrams?
> 
> 
> 


A similar question about the Whitehead tower follows naturally



> 
> **Whitehead tower:** A Whitehead tower of a space is a factorization of the initial morphism $\* \to X$ to a directed limit diagram:
> 
> 
> $$\* \cong\underset{\rightarrow\_n}{\lim}X\_n \to \dots \to X\_2 \to X\_1 \to X\_0 \cong X$$
> Such that $X\_k$ is $(k-1)$-connected and each morphism $X\_{k} \to X\_{k-1}$ is an isomorphism on homotopy groups in degree larger than $k-1$).
> 
> 
> **Question 2:** Does this property determine the Whitehead tower up to weak equivalence of diagrams?
> 
> 
> 



Simple characterization of Postnikov & Whitehead towers?
Indeed, the properties you stated characterize Postnikov and Whitehead towers. A nice conceptual way of justifying this is by using $k$-connected / $k-$truncated factorization systems.


To fix terminology, a $k$-connected map is one whose all homotopy fibers are $k$-connected and a $k$-trunceted map is one whose all homotopy fibers are $k$-truncated. (Note that in the case of connectedness this disagrees with the traditional definition according to which a map is $k$-connected if and only if all its homotopy fibers are $(k-1)$-connected.)


Then every map factors as a composite of a $k$-connected map followed by a $k$-truncated map. Moreover, certain lifting property holds. On the point-set level you could say that $k$-connected cofibrations have the LLP with respect to $k$-truncated fibrations. However, more properly, a homotopy coherent / higher categorical lifting property is satisfied, i.e. for any $k$-connected map $A \to B$ and a $k$-truncated map $X \to Y$ the square of (derived) mapping spaces
$\require{AMScd}$
\begin{CD}
\mathrm{Map}(B, X) @>>> \mathrm{Map}(A, X) \\
@VVV @VVV \\
\mathrm{Map}(B, Y) @>>> \mathrm{Map}(A, Y) \\
\end{CD}
is a homotopy pullback. (In other words, any lifting problem with $A \to B$ on the left and $X \to Y$ on the right has a contractible space of solutions.)


It follows directly that any map factors *uniquely* as a composite of a $k$-connected map followed by a $k$-truncated map, i.e. the space of such factorizations is contractible. Your statements are recovered as special cases of that. The $k$th Postnikov section of a space $X$ is obtained by factoring $X \to \*$ as a composite of a $k$-connected map followed by a $k$-truncated map. In the based case, the $k$th connected cover of $X$ is obtained by factoring $\* \to X$ as a composite of a $(k-1)$-connected map followed by a $(k-1)$-truncated map. (There is an offset here since $X \to \*$ is $k$-connected if and only if $\* \to X$ is $(k-1)$-connected.)


It should also be noted that a map $A \to B$ is $k$-connected if and only if it induces isomorphisms on homotopy groups up to dimension $k$ and an epimorphism in dimension $k + 1$. Similarly, a map $X \to Y$ is $k$-truncated if and only if it induces isomorphisms on homotopy groups above dimension $k + 1$ and a monomorphism in dimension $k + 1$. I believe your description of the Whitehead tower should be adjusted accordingly (although in this particular case, injectivity in dimension $k + 1$ may be vacuous).


