I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.


Suppose you have two diagrams of spaces $X\_{\bullet}, Y\_{\bullet}$ and a homotopy equivalence $f\_i : X\_i \to Y\_i $ such that for any morphism $i\to j$, the morphisms $Y(i\to j)\circ f\_i$ and $f\_j\circ X(i\to j): X\_i\to Y\_j$ are homotopic. Is it true that $\text{holim} X\_{\bullet} \simeq \text{holim} Y\_{\bullet} $ ?


My favorite diagram is $\Delta$, i.e. cosimplicial objects, but I suspect the result should not depend on this. Let me remark that if the $f\_i$ would commute strictly the result would be true.


**Bonus question.** If the answer is "No, they can be different", are there a higher compatibility constraints one can impose so that it becomes true? For example, by specifying what the homotopy $f\_{ij}$ between $Y(i \to j) f\_i $ and $f\_j X(i \to j)$ should respect, and then bla bla..


**Remark**. I suspect that a "not so higher" version should be true for the following reason. If I have a cosimplicial object, I take the chain complex pointwise, and I apply the Dold-Kan correspondence, I get a bicomplex $C\_\*(X\_\bullet)$: the horizontal differential is the differential of chains, while the vertical differential is the alternated sum of the cosimplicial (co)faces.
I have tried to apply the definition of multicomplex homotopy equivalence between $C\_\*(X\_{\bullet})$ and $C\_\*(Y\_{\bullet})$, but there a few more maps to find and I don't want to mess with so many equations. In particular I should find an homotopy inverse to $f$. I hope that the "not so high" is low enough that one homotopy suffices, without higher constraints.



Deformation of a diagram preserve the homotopy limit
This is false.


Consider the two $C\_2$-spaces $S^{2\sigma}$ and $S^2$, where $\sigma$ is the sign representation and $S^V$ denotes the one-point compactification. Then the two underlying spaces are the same and the action looks trivial in the homotopy category. However, the homotopy fixed points differ. (After 2-completion, these look like the same as the actual fixed points- by the Sullivan conjecture- and the fixed points in $S^{2\sigma}$ are $S^0$ which is different than $S^2$.)


The `higher' version would amount to unwinding the definition of a morphism in the $\infty$-category $\mathsf{Fun}(K, \mathcal{C})$ (if you have such a morphism, and it is pointwise an equivalence, then the diagrams are equivalent and hence give equivalent homotopy limits). In the case $\mathcal{C}=\mathsf{Spaces}$, this goes back a ways to work of Dwyer and Kan (see, e.g., their [paper](https://www.jstor.org/stable/2045321?seq=1) "Realizing diagrams in the homotopy category by means of diagrams of simplicial sets.")


In the case of cosimplicial objects in stable $\infty$-categories, we can do better. By the $\infty$-categorical Dold-Kan correspondence, the $\infty$-category of cosimplicial objects is equivalent to the $\infty$-category of cofiltered objects. If $X$ and $Y$ are cofiltered objects, then it really is true that the two diagrams are equivalent you can build equivalences $f\_i$ from $X(i)$ to $Y(i)$ making the diagrams commute up to homotopy. This is because the poset $\mathbb{N}$ can be built as a coequalizer of two maps from $\coprod [1]$ to $\coprod [0]$, so it's easy to build maps out of $\mathbb{N}$.


