The classifying space $B\mathcal{C}$ of a small category $\mathcal{C}$ is by definition the geometric realization of the nerve of $\mathcal{C}$. Now let $\mathcal{C}\_1$ and $\mathcal{C}\_2$ be two small categories. Any functor $F : \mathcal{C}\_1 \rightarrow \mathcal{C}\_2 $ induces a continuous map $BF: B\mathcal{C}\_1 \rightarrow B\mathcal{C}\_2$.


My question is the following: when is a continuous map $f: B\mathcal{C}\_1 \rightarrow B\mathcal{C}\_2$ homotopic to a map of the form $BF$, for some functor $F:\mathcal{C}\_1 \rightarrow \mathcal{C}\_2 $


Thanks.



realization of maps between classifying spaces of categories
Here's a counterexample. Let $M$ be a discrete monoid which is not a group. Consider the 
map
$$
M = \hom (\Bbb N, M) \to \text{maps}\_{\*}(B\Bbb N , BM) = \Omega BM
$$
($\Omega BM =$ the based loops of $BM$ which is the same thing as
Segal's group completion of $M$). 
This map is not a $\pi\_0$ surjection since $M$ isn't a group. 


Now, this fits into the context of your question since $\Bbb N$ and $M$ may be
considered as categories with one object and a functor is just 
a homomorphism (i.e., $\cal C\_1 = \Bbb N$ and ${\cal C}\_2 = M$).


**Remark:** In the case when $\cal C\_1$ and $\cal C\_2$ are both groups, it would appear to me that 
the question you are asking has an affirmative answer.


