Injective objects and retracts of Fra\"iss\'e limits

We present a purely category-theoretic characterization of retracts of Fra"iss'e limits. For this aim, we consider a natural version of injectivity with respect to a pair of categories (a category and its subcategory). It turns out that retracts of Fra"iss'e limits are precisely the objects that are injective relatively to such a pair. One of the applications is a characterization of non-expansive retracts of Urysohn’s universal metric space.