I am using the definition of homotopy limits in terms of ends. That is, for a functor $F:\mathcal{C}\to \textbf{sSet}$ from some category to the category of simplicial sets, the homotopy limit of $F$ is defined to be $$ \operatorname{holim}F = \int_{c\in\mathcal{C}}\textbf{sSet}(N(c\uparrow\mathcal{C}), F(c)). $$ More details about this definition can be found here. By definition of ends, the data of a homotopy limit consists of the object $\operatorname{holim}F$ together with natural maps $$ p_c:\operatorname{holim} F \to \textbf{sSet}(N(c\uparrow\mathcal{C}), F(c)) $$ for each $c \in \mathcal{C}$. Formally, we say that $(\operatorname{holim}F, \{p_c\}_c)$ is a homotopy limit. I would like to prove the following claim.
Claim: Suppose that, for each $c \in \mathcal{C}$, we have a map $p_c':\operatorname{holim}F \to \textbf{sSet}(N(c\uparrow \mathcal{C}), F(c))$ such that $p_c$ is homotopic to $p_c'$. Then $(\operatorname{holim} F, \{p_c'\}_c)$ is also a homotopy limit.
The point here is that I would like to replace the natural maps with homotopic maps, while preserving the property of being a homotopy limit. I think this should be possible, because the whole point of homotopy limits is that they work "modulo homotopy", in the sense that the construction cannot tell the difference between homotopic things. However, I am struggling to prove the claim from my actual definition involving ends. Could anybody help me to understand why this result is true? Ideally I would like a reference where the theorem is actually stated, but an explanation would also be very helpful.