Equivalence of cosimplicial models for homotopy pullbacks

Given a diagram $X_1 \rightarrow X_{12} \leftarrow X_2$ of spaces (though I think the question applies more generally), there are two cosimplicial resolutions which I've seen used to compute the homotopy pullback. The first one, which I'll call $\mathcal A$ is given by $$\mathcal A^p = X_1 \times X_{12}^{\times p} \times X_2$$ and the coface maps are like in a bar (cobar?) construction. This is the one you see, for example, in the construction of the Eilenberg-Moore spectral sequence.

For the other one, let's let $\mathscr P$ denote the pullback category, and $F$ our functor. Then the standard cosimplicial replacement, which I'll write $\mathcal B$, is given by $$\mathcal B^p = \prod_{\substack{x_0 \rightarrow \cdots \rightarrow x_p \\ \in \mathscr P}} F(x_p)$$

It seems that I always encounter one or the other, but not both, which leads me to my question: how do I know they give the same answer? The first one computes the homotopy pullback basically by definition, whereas the second one is a bit mysterious to me. Is there a simple way to compare these two cosimplicial spaces? They do not appear to be level-wise equivalent. Can we say anything about the relationship of their $\mathrm{Tot}$ towers?

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The first one has codegeneracies as well as cofaces, yes? And when you say that it computes the htpy pullback basically by definition I suppose you mean that its Tot is homeomorphic to the space whose points are triples consisting of a point in $X_1$ a point in $X_2$ and a path connecting their images in $X_{12}$? That's how I think of it.
And if you think about Tot of the second one the same way, you should get the space in which a point is a $5$-tuple, one point in each of $X_1$, $X_{12}$, $X_2$ and two paths in $X_{12}$ connecting the middle point with the images of the other two. Which is homeomorphic to the same old thing.
Yes! I think I understand what you are saying better now. The difference between the two can be thought of as just a single subdivision of the path in $X_{12}$. So I wonder if the "canonical equivalence" you mention above can be actually realized as some kind of map induced by this subdivision. I'll give it some thought. Thanks for your help. –  Eric Finster Oct 8 '10 at 12:48