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.