In my lecture course of representation theory of $K$-algebras, we embedded the category of $A$-modules (where $A$ is an associative unital $K$-algebra) into the category of $K<X_i|i\in I>$-modules (where $K<X_i|i\in I>$ is the free $K$-algebra generated by $X_i, i\in I$). Then we studied for a time only the latter category. The embedding was full (i.e. injective on objects, bijective on homsets). But I think it is not always good to study the ''larger category'', since the smaller one can lead us to a much stronger theory for that special case (but not every general). Two good examples would be:
1) Fully embedding the category of fields into the category of commutative unital rings. On one side we get the strong Galois Theory, on the other side we get the usual theory, which is taught in Commutative Algebra (spectrum, Dimension Theory, Modul Theory,...). The striking thing here is, that the theory of Commutative Algebra is in some sense ''trivial for fields'', although it is much more general. Like $Spec(K)=\{(0)\}$ and therefore lots of theorems are trivial for the case of a field.
2) Embedding the category of all vector spaces (not $K$-vector spaces, all vectorspaces! I think this should be a category, if not the concept is the same) into the category of all modules. The first leads to the usual vector space theory with all its power. The latter is much weaker (in some sense), but much more general.
Would you say that it is not always good to look at a more general case (here the larger category) or something like: we should first study the ''easier cases'' well, before attacking the more general one? It is kind of confusing, since we could consider the category of all modules over free algebras, which can be embedded into the category of all modules over algebras. If we consider it like this, then the easier case would be the category of all modules over free algebras and not vice versa (like in the embedding in the first sentence).
Thank you very much for your time!