In all the preceding answers the category $D$ is required to have pushouts / pullbacks in order to epi / mono being equivalent to pointwise epi / mono in functor categories $D^C$. 

The discussion in [On a corollary in Mitchell's book][1] draw my attention to another important class of categories that usually doesn't have pushouts or pullbacks and where epi / mono is also equivalent to pointwise epi / mono in $D^C$: That is when $D$ is exact. 

A category is exact, if 

- it has a zero object 
- kernels and cokernels exist 
- every monomorphism is a kernel and every epimorphism is a cokernel 
- every morphism can be written as a composition of an epimorphism followed by a monomorphism. 

As a reference see Barry Mitchell: "Theory of Categories", II.11 Functor Categories. 


  [1]: https://mathoverflow.net/questions/58010/on-a-corollary-in-mitchells-book