Homotopy Limit is the Limit in the Homotopy Category

Question

I am trying to understand the homotopy limit. This question naturally appears to my mind. Let $I$ be a small category and $\mathcal{X}$ is an $I$-diagram of simplicial sets. There is a functor from the category of simplicial sets to its homotopy category. So we can think $\mathcal{X}$ as an $I$- diagram on the homotopy category of simplicial sets. Then is it true that homotopy limit of the diagram (i.e. limit over the injective fibrant replacement of $\mathcal{X}$) is the limit of the $I$ diagram $\mathcal{X}$ in the homotopy category? In an attempt to prove, for each $i \in I$, we have maps from the homotopy limit of $\mathcal{X}$ to $\mathcal{X}(i)$ in the homotopy category. But however I can't prove the universal property of limit in homotopy category. Please help. Thanks in advance.