The method of forcing in mathematical logic.

If you want to prove the consistency of axiom systems, you can just explicitly present a model of it. To prove results about set theory itself, like the independence of continuum hypothesis or the axiom of choice, you need to find something like a "modified set theory". According to general philosophy, it is enough to present a category, similar to the category of sets. Category theory gives and abundance of such categories: elementary toposes. They are a special class of cartesian closed categories. They come equipped with a natural internal intuitionistic logic, describing the properties of morphisms. Now the problem of finding the required model transforms into the problem of finding a topos, satisfying our set of axioms, interpreted using the internal logic. It can be done categorically. An example of the construction, proving the independence of continuum hypothesis, can be found in P.T. Johnstone's "Topos theory".

Technically one can construct a model in some category of sheaves, but topos-theoretic approach is much more simple and flexible (besides, sheaf is also a categorical notion and any topos is roughly a category of sheaves).

The method of forcing in mathematical logic.

If you want to prove the consistency of axiom systems, you can just explicitly present a model of it. To prove results about set theory itself, like the independence of the continuum hypothesis or the axiom of choice, you need to find something like a "modified set theory". According to general philosophy, it is enough to present a category, similar to the category of sets. Category theory gives an abundance of such categories: elementary toposes. They are a special class of cartesian closed categories. They come equipped with a natural internal intuitionistic logic, describing the properties of morphisms. Now the problem of finding the required model transforms into the problem of finding a topos, satisfying our set of axioms, interpreted using the internal logic. It can be done categorically. An example of the construction, proving the independence of the continuum hypothesis, can be found in P.T. Johnstone's "Topos theory".

Technically one can construct a model in some category of sheaves, but topos-theoretic approach is much more simple and flexible (besides, sheaf is also a categorical notion and any topos is roughly a category of sheaves).