I am working in a model category $\mathcal C$. Given a fibration $p: Y \to B$ and a map $u : A\to B$ where $A$ and $B$ (and thus $Y$ also) are fibrant, it is know that the usual pullback $A\times_B Y$ is an homotopy pullback. I can prove that with an explicit description of the homotopy pullback, no problem.
Do we then have a "universal property" as follow?
For any cofibrant $Z$ (let even assume $Z$ bifibrant if needed), and any maps $f:Z\to Y$, $g:Z \to A$ such that $pf \sim ug$, there exists a unique up to homotopy $k:Z \to A \times_B Y$ such that $g\sim \pi_1 k$ and $f\sim \pi_2k$.
(Where $\pi_1$ and $\pi_2$ are the two projections from the pullback. And $\sim$ stands for "homotopic to"; either right or left, it does not matter here considering the assumptions of (co)fibrancy.)
I am able to prove the existence of such a $k$ (with moreover $g = \pi_1k$ instead of a mere homotopy). It is mostly a use of the homotopy lifting property. But I have trouble proving that two such solutions $k_1,k_2:Z\to A\times_B Y$ are homotopic. Again by the homotopy lifting property I can even suppose that $g = \pi_1k_1 = \pi_1 k_2$ because $\pi_1$ is a fibration (being the pullback of the fibration $p$). So under this hypothesis and the one that $\pi_2k_1 \sim f\sim \pi_2k_2$, I want to deduce that $k_1\sim k_2$.
Any idea? You can even suppose $u$ to be a fibration if it helps. I'm kind of circling around in my attemps and I came to wonder if it needs more assumptions on $\mathcal C$ to be true...
Addendum: there is surprisingly few documents on this "local definition" of an homotopy pullback, compared to the "global definition" as a right derived functor. Appart from the nLab which is kind of vague about the context in which this universal property holds, I did not find anything useful on the web. (Maybe I just lack the correct keywords though...)