Maybe the question is stupid but I can't find a rasonably solution.
Let $\pi:E\rightarrow M$ be a bundle and let $f:N\rightarrow M$ be a map bewteen manifolds. The fibred product bundle is the bundle whose total space is the set
$$ E\times_M N = \{(x,p)\in E\times N | \pi(x)=f(p)\} \subset E\times N . $$
But now, in some texts, Riemannian metrics on vector bundles are defined as vector bundle morphisms
$$ g:E\times_M E\longrightarrow M\times \mathbb K $$ such that in fibres they are inner products (here $\mathbb K$ means the field over the fibre is defined).
The question is: Who is this fibred product? Which map (or maps) should I use to define it? I thought in local trivializations, but that doesn't make sense. I don't know...
PD: I'm thinking in vector bundles because my question arised with vector bundles. But the answer can be done perfectly with sets. The problem is the fibred product of some set with itself.