Is this statement (about the localization of a category) true? Say we are localizing a category $\mathcal{B}$ at the $P$-morphisms, where by $P$-morphism I mean a morphism having some property $P$. 
Fix a category $\mathcal{C}$. Is it true that the category of functors from the resulting category of fractions of $\mathcal{B}$ to $\mathcal{C}$ is isomorphic to the category of functors from $\mathcal{B}$ to $\mathcal{C}$ that turn $P$-morphisms to isomorphisms? 

As Zhen suggested: the definition of a localization of $\mathcal C$ at $S$ is that the functor categories $[\mathcal C[S^{-1}],\mathcal D]$ are isomorphic to the categories of functors $[\mathcal C,\mathcal D]$ sending $S$ to isomorphisms, via composition with the localization functor. 
