Take the following small model for the category of finite-dimensional vector spaces and isomorphisms: The set of objects is $\mathbb{N}$ and the set of morphisms $Mor(n,m)$ is empty, if $n \neq m$ and $U(n)$ otherwise. If we take the classifying space of that, we get $\coprod_{n \in \mathbb{N}} BU(n)$.

What happens, if we take $Mor(n,m) = Emb(\mathbb{C}^n, \mathbb{C}^m)$, where the latter denotes the space of isometric linear maps instead, i.e. what is the homotopy type of the corresponding classifying space?

Note that $\mathbb{N}$ does not contain zero, so there is a priori no reason why this should be contractible. Moreover, there is a map from $\coprod_{n \in \mathbb{N}} BU(n)$ into the space described above, induced by the canonical functor. Nevertheless, it is connected, since there is always a canonical embedding $\mathbb{C}^n \to \mathbb{C}^m$ for $m \geq n$.

So, my guess would be that this is a model for $BU$, since it somehow looks like some kind of telescope construction. But I could not prove this, because there is no canonical extension of an embedding $\mathbb{C}^n \to \mathbb{C}^m$ to a unitary. So, am I right or is this just another elaborate description of the point?