I have a concrete problem with the homotopy fiber and I am getting lost with the literature. I state my question and, to avoid confusions, I state downwards the definitions I am using.

Let $C$ be a pointed model category. To make things simpler, assume every object is fibrant. Recall that the definition of the homotopy fiber of a morphism $f\colon Y\to X$ is the homotopy pullback of the diagram $$\begin{matrix}& & Y \\ &&\downarrow^f \\ *&\to & X\end{matrix} $$ That is to say, we replace $*\to X$ and $f$ by fibrations (in the category of diagrams) and take the classic pullback.

Now, if I have a map $Z\to X$ that fits in a commutative square $$ \begin{matrix} Z&\to &Y\\ \downarrow &&\downarrow^f\\ *&\to & X. \end{matrix}\quad (1) $$ then it factors through the fiber. More concretely, there is a map $Z\to \mathrm{fib}$ such that the diagram $$ \begin{matrix} Z&\to&\mathrm{fib}&\to &Y\\ &&\downarrow &&\downarrow^f\\ &&*&\to & X. \end{matrix} $$ commutes in $C$ and the composed maps $Z\to *$ and $Z\to X$ are those of (1).

My question: **is there an analogue map to the homotopy
fiber?**, i. e., is there a commutative square in $C$
$$
\begin{matrix}
Z&\to&\mathrm{hofib}&\to &Y'\\
&&\downarrow &&\downarrow^{Rf}\\
&&*'&\to & X'.
\end{matrix}\quad (2)
$$
where $*'\to X'$ and $Rf\colon Y'\to X'$ are the fibrant replacements?

Let me recall the general definition of the homotopy limit. Denote $D$ the above
diagram. Then $C^D$ has a model structure. Note $\mathrm{lim}\colon C^D\to C$
the functor *taking pullback*. The homotopy pullback is the right derived
functor of $\mathrm{lim}$ in the sense of Quillen in "Homotopical algebra".
Therefore, in order to have such a map I should have a functor $F$ such that for
*every* diagram $d\colon D\to C$ I have an object $F(d)$ with maps as in (1) and
that in my concrete diagram gives $Z$. Then I would have a diagram like (2). Right?

I have read in n-category lab that this definition is called *global* and there is another definition of the homotopy fiber called *local* and that, under certein hypothesis, they seem coincide. Do this terminology refer to this problem? I am unable to understand the notations in the literature.