Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have $$d(x,y)\ge d'(fx,fy)\,.$$ Let $\mathbf{Met}_{\not=\emptyset}$ be the full subcategory of $\mathbf{Met}$ with non-empty spaces as objects.
The category $\mathbf{Met}$ is, what Wikipedia defines as the category of metric spaces. However, many Analysis books require metric spaces to be non-empty.
My question is, which of the above two categories is more well-behaved, and which definition should therefore be preferred from a categorical point of view.
For example, $\mathbf{Met}_{\not=\emptyset}$ has no initial object, whereas $\mathbf{Met}$ does, namely the empty space which uniquely embeds into every other metric space. Are there other differences concerning existence of limits and colimits? Are arrows in $\mathbf{Met}_{\not=\emptyset}$ (split/effective/descent/regular/extremal) monomorphisms and epimorphisms iff their images under the embedding functor to $\mathbf{Met}$ are, or do the characterizations change?
Note that this question is not a duplicate of SE/45145, since I am not merely interested in topological but rather in categorical aspects.
