*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?